Introduction

• In these notes, unless otherwise stated, all sets are considered to be proper sets as in ZFC.
• Basic (at least naive) set theory and simple combinatorics knowledge is assumed. Other than that, notes should be self-sufficient.

At the moment, these notes have a formal and reference-book like approach except these grayed out notes. I plan more intuition baked in for these notes, with much more visuals, examples and geometry involved.

Resources Used

• Algebra by Thomas W. Hungerford
• Fundamentals of the Theory of Groups, translated second Russian Ed., by M.I. Kargapolov and Ju.I. Merzljakov
• Group Theory Exercises and Solutions by Mahmut Kuzucuoğlu
• Graduate Algebra Problems with Solutions by Mahmut Kuzucuoğlu
• Abstract Algebra, 3rd Ed., by David S. Dummit and Richard M. Foote

Notation

• 0 \in \N and \N^+ :=\ \N \setminus \{0\}.
• \varnothing denotes the empty set.
• (m,n) denotes the greatest common divisor of m, n \in \N.
• \equiv_m denotes integer equivalence in modulo m.
• Cardinality of a set S is denoted with |S|.
• d_1 \>|\>d_2\>|\>\cdots\>|\>d_r means d_1 divides d_2 divides d_3 etc.

Status

These notes are still under work therefore not complete, also

• there are not many exercises,
• proofs are mostly absent,
• typos are quite possible,
• ordering is generally good but should be improved, and
• more visuals and intuition should be provided.

Def. Group

A group is an ordered pair (G, \cdot) where G is a set and \cdot is a binary operation on G that satisfies:

Simply, \cdot is a (total) function from G to G. Notice that G is any set, finite or infinite.

• Associativity, that is, for all a,b,c \in G we have (a \cdot b) \cdot c = a \cdot (b \cdot c)

This alone defines a semigroup.

• Identity, that is, there exists e \in G called identity (of G) such that for all a \in G we have a \cdot e = e \cdot a = a.

Until here it defines a monoid where identity is two-sided, namely left and right.

• Inverse, that is, for each a \in G there exists an element (called inverse) b \in G such that a \cdot b = b \cdot a = e.

Noting that the identity of a group and the inverse of an element in that group is always unique (exercise) we will denote the inverse of an element a with a^{-1} unless it is abelian.

A group is called abelian (or commutative) if its elements commute, that is, if for all a,b \in G we have a \cdot b = b \cdot a. For abelian groups, we may prefer the additive notation + instead of \cdot for the binary operation and denote the inverse with -a instead.

We might also consider the group as a triplet with identiy (G,\cdot, e) as it is not clear otherwise what is the identity explicitly.

Remarks

The definition (or axioms) given above are not minimal. For example, it's enough to just accept right-identity and right-inverse for it to be group. Using just these two, you can later prove it also holds for the left-identity and left-inverse with the help of the associative property.

Associative property by far is the most powerful property of the group. It allows you to write your expression (involving only \cdot) without any parentheses and much more.

Indeed a structure which only satisfies associative property is called a semigroup. A semigroup with identity is called a monoid and a monoid with inverses is called a group.

Thm. Basic Group Properties

Remembering any group is also a monoid and thus a semigroup let (G, \cdot) be a group, then:

1. Identity e is unique. The uniqueness of the identity element does not require the use of associativity.
2. For each a \in G, inverse of a is unique.
3. For each a \in G, we have (a^{-1})^{-1} = a.
4. For all a,b \in G, we have (a \cdot b)^{-1} = b^{-1} \cdot a^{-1}. Indeed, in general, (a_1 \cdots a_n)^{-1} = a_n^{-1} \cdots a_1^{-1}.

Exercise

Thm. Basic Monoid Properties

If M is a monoid, then

1. The identity element of M is unique.

Thm. Semigroup to Group

Let (S, \cdot) be a semigroup, then it is a group if and only if both of the following hold:

• Left-identity exists, and
• Left-inverse exists for each s \in S.

By symmetry, the analogous result holds for rights instead of left.

Thm. Semigroup to Group 2

Let S be a semigroup, then it is a group if and only if for all a,b \in S the equations

have solutions in G.

Exercise

Thm. Generalized Associative Law

Let S be a semigroup and a_i \in S. Associative property implies that the expression a_1 \cdot a_2 \cdot \cdots \cdot a_n is the same no matter how the expression bracketed.

Similarly one could also prove Generalized Commutative Law for the commutative property.

Def. Order

Let (G, \cdot) be a group and a \in G.

The order of (the group) G is denoted by |G| and is the cardinality of the set G.

The order of (the element) a is denoted by |a| and (if exists) it is the least positive integer n such that x^n = e. If there is no such n, we say the order is infinite.

Order of an element a is sometimes denoted with o(a).

If the order of an element x (or group) is finite, we will denote it with |x| \lt \infty. Moreover, if x^2 = x, then x is called an idempotent element where e is the trivial idempotent element.

We say that a group if torsion-free if every non-identity element has infinite order. If every element of a group has finite order then we say the group is periodic.

If orders of a periodic group are bounded, then the least common multiple of their orders is called the exponent of the group. If the orders of elements of a periodic group are powers of prime p, then we call the group a p-group.

Notation. Subsets

Let G be a group and A,B \subseteq G, then we define

1. AB := \Set{ab \in G | a \in A, \> b \in B},
2. A^0 := \{e\},
3. A^n := AA^{n-1},
4. A^{-1} := \Set{a^{-1} \in G| a \in A}.

Notation. The Additive Notation

If the binary operation is written additively, which is mostly the case for abelian groups, we may write:

• 0 for the identity instead of 1 (or e for that matter).
• n a instead of a^n where n \in \Z. Notice that operation between n and a is not the binary operation of our structure but rather "n times a".

We define a^0 (or 0a) as the identity element 1 or 0. Notice that, in additive notation, 0a is not the multiplication by the identity but rather "0 times n" which we define to be the identity 0.

Thm. More Group Properties

Let G be a group, then

1. If a^2 = e for all a \in G, then G is abelian. (Such groups are called elementary abelian 2-groups.)
2. If |G| is finite and even, then it has an element x of order 2. Moreover, x \in Z(G) that is g^{-1}xg = x for all g \in G.
3. If A \subseteq G and g \in G, then |A| = |gA|=|Ag|.

Exercises

Klein 4-Group

See Wikipedia: Klein four-group.

The Klein 4-group can be defined by the group presentation

Such group is

• of order 4,
• Abelian,
• all non-identity elements have order 2,
• smallest non-cyclic group,
• isomorphic to Dihedral Group of order 4,
• isomorphic to \Z_2 \oplus \Z_2.

Also note that any group of order 4 is isomorphic to either \Z_4 or \Z_2 \oplus \Z_2.

Dihedral Groups

See Wikipedia: Dihedral group.

Symmetric Groups

See Wikipedia: Dihedral group.

The Quaternion Group

See Wikipedia: Quaternion group.

The \mathbb{Q}_p Group

For p prime, define

so that \mathbb{Q}_p is a (torsion-free) abelian group under the usual rational addition.

Exercises

Def. Subgroup

Let G be a group and non-empty H \subseteq G. The non-empty subset H is called a subgroup if H is again a group under the restriction of G's binary operation. This implies H has the same identity as G under the same binary operation.

Thm. Equivalent Subgroup Definitions

A subset H \subseteq G is a subgroup of G if

• H has the same identity as G,
• For all a,b \in H, we have ab \in H that is HH \subseteq H,
• Every element h \in H has an inverse that is H^{-1} \subseteq H.

To be more compact, non-empty H \subseteq G is called a subgroup if and only if (exercise):

• For all a,b \in H we have ab^{-1} \in H.

From now on, we will denote by H \leq G that H is a subgroup of G, moreover H \lt G if H \neq G. The latter is called a proper subgroup of G.

Any group has two subgroups called the trivial subgroup which consists of only the identity and the group itself.

Convention regarding to this trivial and proper notation differs from author to author — we will stick to this naming.

Example. Some Subgroups

• Under addition, \Z \lt \mathbb{Q}_p, \lt \mathbb{Q} \lt \R \lt \Complex,
• Under addition, \Z = \bigcap \mathbb{Q}_p,
• \mathbf{GF}(p^m) \leq \mathbf{GF}(p^n) if m \mid n where \mathbf{GF}(p^m) is the appropriate subset of the algebraic closure of \mathbf{GF}(p).

• Under multiplication, \Z^* \lt \mathbb{Q}^*, \lt \R^* \lt \Complex^*,
• Under multiplication, \Complex_p^* \lt \Complex_{p^2} \lt \cdots \lt \Complex_{p^\infty},
• \Complex_{p^\infty} = \bigcup \Complex_{p^n},
• \mathbf{GF}(p^m)^* \leq \mathbf{GF}(p^n)^* if m \mid n.

• The subset A_n of all even permutations forms a subgroup called the alternating group of degree n, and |A_n|=n!/2.

Thm. Finite and Closed Subset

Let G be a group and S a non-empty subset of G. If S is finite and closed under the group product, then S is a subgroup of G.

So, we don't even need the inverse condition if S non-empty and finite.

Thm. Intersection of Subgroups

Let \{H_i\} be any non-empty family of subgroups of G, then \bigcap H_i is also a subgroup of G.

Exercise

Thm. Subgroups Under Multiplication

Let G be a group and H,K \leq G, then

• HH = H and H^{-1} = H, thus obviously
• HH^{-1} = H,
• HK is a subgroup of G if and only if HK=KH, and

Exercise

Def. Complement

Let H \leq G. We say K is a complement of a subgroup H if

• G = HK, and
• H \cap K = \{e\}.

Noting KH=HK, this complement relation is symmetrical.

Thm. Basic Complement Properties

1. Complements need not to exists, and if they exists they need not to be unique.

Let H and K be complements in G, then

2. Every element of G has an unique expression as a product hk or k'h' where h,h' \in H and k,k' \in K.
3. K forms both left and right transversal of H for the cosets of H.

Def. Maximal Subgroup

Let G be a group and let H be a proper subgroups of G. We say H is a maximal subgroup if H \subseteq K implies K = H for all K \lt G.

Simply, H is maximal if there is no greater proper subgroup which contain it.

Def. Frattini Subgroup

Let G be a group. We define frattini subgroup \Phi(G) as the intersection of all maximal subgroups of G. In the case G has no maximal subgroups, we define \Phi(G) = G.

This is analogous to the Jacobson radical in the ring theory.

Thm. Frattini Subgroup and Non-Generators

The frattini subgroup \Phi(G) of a group G is equal to the set of all non-generators of G. Therefore, non-generators of a group form a subgroup — namely the frattini subgroup.

Exercises

Def. Homomorphism

Let (G, \cdot_G) and (H, \cdot_H) be semigroups.

The (total) function (or map) \varphi: G \to H is called a homomorphism if, for all a, b \in G:

Mostly, we will not be as explicit about the operations and simply write \varphi(ab)=\varphi(a)\varphi(b) instead.

The homomorphism \varphi is called:

• an monomorphism if it is injective,
• an epimorphism if it is surjective,
• an isomorphism if it is bijective.

• an endomorphism if G=H, and
• an automorphism if it is an endomorphism and bijective.

Composition of homomorphisms is again a homomorphism. Respectively, this is also the case for monomorphisms, epimorphisms, isomorphisms and automorphisms.

For example, if A is abelian, then the map a \mapsto a^{-1} is an automorphism, and the map a \mapsto a^2 is an endomorphism.

Def. Kernel

If \varphi: G \to H is a group homomorphism, then the kernel of \varphi is defined as

Notation. Homomorphisms

We say semigroups G and H are isomorphic denoted with G \cong H if there exists an isomorphism between them.

Let \phi: G \to H be a group homomorphism g \in G and A \subseteq G, then

• g^\phi denotes \phi(g), and
• A^\phi denotes \phi(A) called the homomorphic (respectively monomorphic, epimorphic, ...) image of A.

Thm. Basic Homomorphism Properties

Let \varphi: G \to H be a group homomorphism, then

1. \varphi(e_G) = e_H. This is not necessarily true for monoid homomorphisms!
2. \varphi(g^{-1}) = \varphi(g)^{-1} for all g \in G,
3. \varphi(g^n) = \varphi(g)^n for all g \in G and n \in \Z,
4. \text{Ker }\varphi \leq G,
5. \text{Im }\varphi := \varphi(G) \leq H

Exercise

Def. Basic Kernel Properties

Let \varphi: G \to H be a group homomorphism, then

1. \varphi is a monomorphism if and only if \text{Ker } \varphi = \{e_G\}.
2. \varphi is an isomorphism if and only if there exists an homomorphism \varphi^{-1}: H \to G such that \varphi \varphi^{-1} = \text{id}_G.

Exercise

Def. Endomorphisms

Let G be a group and \text{End } G the set of all endomorphism on G, then \text{End } G is a semigroup under composition. Moreover, if G is a abelian, \text{End } G is a ring with pointwise function addition that is, for \alpha, \beta \in \text{End } G

Exercise

Def. Automorphisms

The set of automorphisms on G denoted by \text{Aut }G is a group under function composition. Moreover, \text{Aut }G \leq \mathbf{S}(G) where \mathbf{S}(G) is the group of permutations on G.

Exercise

Exercises

Def. Generators

From now on, for a group G and a subset A \subseteq G, we will denote by L(G, A) the set of all subgroups of G that contain A. In particular, L(G) denotes the set of all subgroups of G.

Noting that intersection of any collection of subgroups are again a subgroup, we define for any set M \subseteq G, the subgroup generated by M, denoted \Braket{M}, as the intersection of all subgroups which contain M. That is

Elements of M are called the generators of the subgroup \Braket{M}. If M is finite, then we say \Braket{M} is finitely generated.

From now on, when we use set builder notation, instead of \Braket{\Set{x_1, x_2, ... \in X | \cdots}} we will omit the parentheses and simply write \Braket{x_1, x_2, ... | \cdots}.

An element is called a non-generator of a group G if it can be omitted from every generating set for G.

Generally, this definition of a generated subgroup is not really easy to work with. So equivalently...

Thm. Equivalent Generation Definition 1

If M is a subset of a group G, then

Thm. Equivalent Generation Definition 2

Let G be a group and M \subseteq G, then

That is, \Braket{M} consists of all finite products of a_1^{n_1} \cdots a_k^{n_k}.

Therefore, in particular \Braket{x} = \Set{x^n | n \in \Z}. We will inspect these structures in detail in the next chapter.

Def. Join of Subgroups

Let H_i be subgroups of G, then their join is defined as \Braket{\> \bigcup H_i \>} or, if finitely many, as \Braket{H_1, ..., H_n}. The join of two subgroups H,K will simply be denoted as H \lor K.

This notation will make sense later on when we define lattices over groups.

Example. Generator Examples

• \Z = \Braket{1},
• \Z_n = \Braket{\bar{1}},
• \mathbb{Q} = \Braket{\dfrac{1}{n} \mid n = 1,2,\dots},

• \Z^* = \Braket{-1},
• \mathbb{Q}^* = \Braket{-1, 2, 3, 5, 11, \dots},

Def. Cyclic Group

A group H is called cyclic group, or simply cyclic, if H can be generated by a single element. That is, there exists an element x \in H such that H = \Braket{x} = \Set{x^n \mid n \in \Z}. Such x is called the generator of H or H is generated by x.

Since cyclic groups are abelian (exercise), additive notation may also be used. In that case, x^n becomes nx.

Notice that the order of the element x and the group \Braket{x} are the same.

Thm. Basic Element Order Properties

Let G be any group and a \in G, then

In the case |a| is not finite,

1. a^k = e if and only if k = 0,
2. each a^k is distinct for k \in \Z.

In the case |a| = n \in \N^+,

3. n is the least positive integer such that a^n = e,
4. a^k = e if and only if n\>|\>k,
5. a^r = a^s if and only if r \equiv_n s,
6. for each k\>|\>n, we have |a^k| = \frac{n}{k}.

Exercise

Thm. Basic Cyclic Properties

Let H be a cyclic group, then

• H is also abelian. So, cyclic implies abelian!
• If x is a generator of H, then so is x^{-1}.
• If x is a generator of H, then |H| = |x|.

Thm. Fundamental Order Property

Let G be a group, g \in G, and m,n \in \Z. If x^m = e and x^n = e, then x^d = e where d=(m,n).

In particular, for any m such that x^m = e, we have |x| divides m.

Thm. Every Subgroup of \Z is Also Cyclic

Noting subgroup of a cyclic is cyclic, let (H, +) \leq (\Z, +). Then, either

• H=\Braket{0} which is the trivial subgroup \{0\}, or
• H=\Braket{m} where m is the least positive integer in H. In this case, H is infinite.

Thm. Same Order Cyclics are Isomorphic

For any two cyclic groups \Braket{x} and \Braket{y}, if their orders are the same, there exists an isomorphism \varphi : \Braket{x} \to \Braket{y}.

1. Indeed, if they are finite, then the map

   \def\arraystretch{1.5} \begin{array}{cc} \varphi: & \Braket{x} \to \Braket{y} \\ & x^k \mapsto y^k \end{array}

   is well-defined and an isomorphism. Therefore, any finite cyclic group of order n is isomorphic to the cyclic group (\Z_n, +_Z).

2. If they are infinite, then the map

   \def\arraystretch{1.5} \begin{array}{cc} \varphi: & \Z \to \Braket{x} \\ & k \mapsto x^k \end{array}

   is well-defined and an isomorphism. Therefore, any infinite cyclic group is isomorphic to (\Z, +_\Z).

Thm. Fundamentals of Element Orders

Let G be any group, x \in G and a \in \Z^*, then

1. If |x| = \infty, then |x^a|=\infty.
2. If |x| = n, then |x^a| = \dfrac{n}{(n, a)}.

Thm. Orders of Commutative Elements

Let G be a group and a and b elements of G whose orders are respectively m and n. If a and b commute, then

1. (m,n) = 1 \implies |ab| = |a| \> |b|,
2. There exists g \in G such that |g| = \text{lcm}(m,n).

Thm. On Generators of Cyclics

Let H = \Braket{x}, then

1. If H is infinite, then x and x^{-1} are the only generators of H.
2. If H is finite of order n, then x^k is a generator of H, if and only if (k,n)=1.

Therefore, the number of generators of H equals to \varphi(n) where \varphi is Euler's \phi-function.

Thm. Basic Cyclic Properties

Let H=\Braket{x} be cyclic, then

1. Every subgroup of H is also cyclic.
2. If H is infinite, then for any distinct non-negative integers a and b, \Braket{x^a} \neq \Braket{x^b}.
3. For every integer m we have \Braket{x^m} = \Braket{x^{-m}}. Therefore, every non-trivial subgroup of H...

Thm. Homomorphisms from Cyclics

Let G = \Braket{a} be a cyclic group and H any group, then every homomorphism \varphi: G \to H is completely determined by the element \varphi(a) \in H. In particular, \text{Im }\varphi = \Braket{\varphi(a)}.

Obvious

Thm. Finitely Many Subgroups Imply Finite Group

Any group which has only finitely many subgroups must also be finite.

Exercises

Def. Coset

Let G be a group and H \leq G. Then, for all a \in G the set aH is called a left coset and the set Ha is called a right coset.

Def. Coset Congruence

Let G be a group, H \leq G, and a,b \in G. We say,

• a is left-congruent to b modulo H, denoted by a \equiv_{L} b \enspace (\text{mod } H) when a^{-1}b \in H.
• a is right-congruent to b modulo H, denoted by a \equiv_{R} b \enspace (\text{mod } H) when ab^{-1} \in H,

Thm. Coset Congruence

1. The relations \equiv_L and \equiv_R are equivalence relations.
2. The left (resp. right) equivalence class of a \in G is the set aH (resp. Ha).
3. For all a \in G, cardinalities of the sets Ha, H and aH are the same.
4. If G is abelian, then left and right congruence coincide. Moreover, this is also possible if G is not abelian.

Def. Index

Let G be group and H \leq G then the index of H in G, denoted |G:H| is the cardinal number of the set of distinct right (or left) cosets of H in G.

Thm. Index Theorem

Let G be a group and K \leq H \leq G, then

Thm. Cauchy's Theorem

Let G be a finite group of order n and p is any prime that divides n. Then G contains an element of order p.

We will prove this useful theorem later on, after Sylow Theorems.

Thm. Poincaré

Every subgroup of finite index m of a group G contains a normal subgroup of G of finite index n divisible by m. Moreover, n divides m!.

Thm. Order of Subgroup Multiplication

Let G be group such that H and K are finite subgroups of G, then

Thm. 1

Let G be a group and H,K \leq G, then we have |H:(H \cap K)| \leq |G:K|.

Moreover, if |G:K| is finite, then |H:(H \cap K)|=|G:K| if and only if G = KH.

Thm. 2

Let H and K be subgroups of finite index of a group G, then

1. |G:H \cap K| is finite,
2. |G:H \cap K| \leq |G:H||G:K|, and
3. |G:H \cap K| = |G:H||G:K| if and only if G = HK.

Thm. Groups of Prime Order

Let G be a group, then the following are equivalent

1. |G| is prime,
2. G \neq \Braket{e} and G has no proper subgroups,
3. G \cong \Z_p for some prime p.

Notice that (3) implies that every group of prime order is cyclic.

Exercises

Def. Conjugate

Let G be a group, H \leq G, and a,b \in G, then

1. the element aba^{-1} is called the conjugate of a by b,
2. the set aHa^{-1} is called the conjugate of H by a,
3. the element a is said to normalize H if aHa^{-1} = H.

Note that more general definitions would use only commutativity (that is gh = hg) instead of inverses for semigroups.

We also say a is conjugate to an element b by an element x if a = xbx^{-1} denoted with a = b^x. We further define for sets A, B \subseteq G, and g \in G

Note that A^B is defined as the set of elements bab^{-1}, not ba(b')^{-1} for some b'.

Thm. Basic Conjugate Properties

Let G a group and a,b,x \in G, then

• (ab)^x = a^x b^x,
• (a^x)^y = a^{xy},
• a=b^x \implies |a| = |b|.

Def. Normal

Let G be a group and N its subgroup. If for all a \in G we have aN=Na, then N is called a normal subgroup (or simply a normal) of G denoted by N \trianglelefteq G.

If N \neq G, then N \vartriangleleft G will also be used to denote N is a proper normal subgroup of G.

From now on, it should be understood from A \trianglelefteq B alone that B is a group and A is its normal subgroup.

Thm. Equivalent Normal Definitions

Let G be a group and N \leq G, then the following are equivalent

1. \equiv_L and \equiv_R modulo N coincide,
2. gN=Ng,
3. N^g = gNg^{-1} \subseteq N for all g \in G, that is N^G \subseteq N,
4. N^g = gNg^{-1} = N for all g \in G, that is N^G = N.

Thm. More Normal Properties

1. Let M,N \trianglelefteq G. If M \cap N = \{e\}, then mn=nm for all m \in M and n \in N.
2. Kernel of any group homomorphism is a normal subgroup.
3. If |G:H| = 2, then H \trianglelefteq G.
4. A, B \trianglelefteq G implies AB \trianglelefteq G.
5. Find normal subgroups A, B, C such that A \trianglelefteq B \trianglelefteq C, but A \not\trianglelefteq C.

Thm. Normal and Subgroup Properties

Recall that the "join" of two subgroup H,K denoted H \lor K is the subgroup \Braket{H \cup K}.

Let N \trianglelefteq G and K \leq G, then

1. (N \cap K) \trianglelefteq G, so intersection of any subgroup with a normal is a normal,
2. N \lor K = NK = KN, so join of any subgroup with a normal is their product,
3. N \trianglelefteq (N \lor K).

TODO: Revise (2) noting that we have defined the multiplication as join! Did we define that?

Def. Simple Group

A group is said to be simple if it has no proper (and non-trivial) normal subgroups.

Exercises

Def. Centralizer

Let G be a (sub)group and A a non-empty subset of G, then the centralizer of A in a group G is defined as

Beware that if we were to write A^g = A to the right-hand side it wouldn't be the same definition.

Note that a more general definition would use ga = ag for semigroups.

Def. Center

The center of a (sub)group G denoted with Z(G) is defined as Z(G) := C_G(G).

It is basically the set of all elements in the group that commute with all other elements in the group.

Def. Normalizer

Let G be a group and A a non-empty subset of G. Similar to centralizer (but not necessarily equivalent), the normalizer of A in G is defined as

and it is also a subgroup of G.

The definitions of centralizer and normalizer are similar but not identical. If g \in C_G(A) and a \in A, then it must be the case that a^g = a, but if g \in N_G(S), then a^g = a' for some a' \in A, with a' possibly different from a.

Obviously a subgroup is a normal subgroup in a group if and only if its normalizer is the whole group.

This is one reason why the notation gag^{-1} (or a^g) is preferred over ga=ag — unless we working with semigroups of course.

Thm. Basic Properties of Normalizer and Centralizer

Let G be a group, then

1. Z(G) \trianglelefteq G

Thm. '

TODO: Revise, define a^G etc.

Let G be a group and a \in G, then

You may check out Kargapolov p. 16 for a more general version of theorem and the proof.

Notation. Normal Generators

Let H \leq G, then

• H^G denotes the intersection all normals in G that contain H,
• H_G denotes \Braket{H^g | g \in G}.

join and largest normal subgroup contained in H.

Thm. Building Normal from a Subgroup

Let H \leq G, then the set

is a normal subgroup of G. Moreover, N = H_G.

Exercise

Exercise

If G is not abelian, then Z(G) is properly contained in an abelian subgroup of G.

Def. (Group) Congruence Relation

An equivalence relation \equiv on a group G is called a (group) congruence relation if for all x_1,x_2,y_1,y_2 \in G

The product of two congruence classes is again a congruence class. Indeed, the set of all congruence classes G/{\equiv} is a group under the multiplication of classes called the quotient group with respect to \equiv.

Thm. Group Congruences and Normals

The congruence relations on a group G are in one-to-one correspondence with the normal subgroups of G.

Usually quotient groups in group theory are defined via normal groups but this paints a much wider picture. Following this motivation, here is the classical definition of quotient groups.

Def. Quotient Group

Let G be a group and N \trianglelefteq G. The set of all cosets of N in G denoted by G/N (read as G modulo N) forms a group under the binary operation

and is of order [G:N]. This group is called the quotient group (or factor group) of G by N.

Notice how we are not multiplying cosets directly, but rather the elements in front of them.

Thm. Basic Quotient Properties

Let G be a group and N \trianglelefteq G.

1. If G is cyclic, then so is G/N.
2. G/N is abelian if and only if [G,G] \subseteq N.

Exercise

Def. Projection

Let N \trianglelefteq G, then

is an epimorphism and \text{Ker }\pi = N. Such \pi is called the canonical epimorphism or (natural) projection of G under N. Therefore, unless otherwise stated, G \to G/N always denotes the natural projection.

If the group is clear from the context, we may make use of the notation \pi_N to denote the projection G \to G/N.

Exercise

Thm. Commutativity of Projection

Let \pi_N be the natural projection of G under N, then G/N is abelian if and only if [G,G] \subseteq N.

Thm. Fundamental Theorem on Homomorphisms

Let \varphi: G \to H be a group homomorphism, N \trianglelefteq G, and N \subseteq \text{Ker }\varphi \trianglelefteq G. Then there exists an unique homomorphism \bar{\varphi} where

and

• \varphi(G) = \bar{\varphi}(G/N),
• \text{Ker }\bar{\varphi} = (\text{Ker }\varphi) / N

Therefore, \bar{\varphi} is an isomorphism if and only if

• \varphi is an epimorphism, and
• N = \text{Ker }\varphi.

Thm. First Isomorphism Theorem

Let \varphi: G \to H be a group homomorphism, then

1. \text{Ker }\varphi \trianglelefteq G, so kernel of any group homomorphism is normal,
2. \varphi(G) \leq H, so image of any group homomorphism is a subgroup,
3. \varphi(G) \cong G/(\text{Ker }\varphi), so if \varphi is an epimorphism, then H \cong G/(\text{Ker }\varphi).

Figure 1. First Isomorphism Theorem

Thm. Second Isomorphism Theorem

This theorem is also called the Diamond Isomorphism Theorem or Parallelogram Theorem due to lattice it draws.

Let G be a group H \leq G and N \trianglelefteq G, then

Recall that since N is normal and H is a subgroup, we have H \lor N = HN = NH.

1. N \trianglelefteq HN \leq G,
2. H \cap N \trianglelefteq H, and
3. HN/N \cong H/(H \cap N).

TODO: (Examine) Technically, N need not to be normal in G, it suffices H to be a subgroup of N_G(N).

Figure 2. Second Isomorphism Theorem

TODO: Redraw diagram

Thm. Third Isomorphism Theorem

Let K \trianglelefteq H \trianglelefteq G, then

1. H/K \trianglelefteq G/K, and
2. (G/K)/(H/K) \cong G/H.

Thm. Homomorphism Induced Bijection

Recall that L(G,A) was the set of all subgroups of G which contain the subset A, and L(G) := L(G, e).

Let \varphi: G \to H be a group homomorphism. Then \varphi induces a bijective map

such that image of normal subgroups are normal subgroups.

TODO: Proof, omitted.

Corollary. Normal Subgroups of Qutients

Let N \trianglelefteq G, then every subgroup of G/N is of the form K/N where N \subseteq K \leq G. Moreover, K/N \trianglelefteq G/N if and only if K \trianglelefteq G.

Def. Inner and Outer Automorphisms

Let G be a group, a \in G, and \iota_a: G \to G be a map such that x \mapsto x^a, then \iota_a is an automorphism on G called an inner automorphism. Moreover, the set of all inner automorphism on G denoted by \text{Inn }G is a normal subgroup of \text{Aut }G.

An automorphism which is not inner is called an outer automorphism. Noting \text{Inn }G is normal, we define the outer automorphism group as

Thm. Inner Automorphisms

Let G be a group, then

Def. Endomorphic Invariance

Let H \leq G and \Phi \subseteq \text{End }G. We say H is \Phi-invariant or invariant with respect to \Phi if for all \phi \in \Phi

Noting that \Braket{e} and G is invariant with respect to any arbitrary \Phi, we say the group is \Phi-simple if it contains no other \Phi-invariants than these two.

Thm. Equivalent Normal Definition

Let H \leq G, then H is normal if and only if H is invariant with respect to \text{Inn }G.

Exercise

Notation. Invariance

Let H \leq G, then we respectively denote \text{End } G, \text{Aut }G, and \text{Inn }G with

• \leq_E or \leq_\text{End},
• \leq_A or \leq_\text{Aut},
• \leq_I or \leq_\text{Inn} which is equivalent to \trianglelefteq as shown above.

Thm. Invariance Transitivity

The relations \leq_\text{End} and \leq_\text{Aut} are transitive.

Exercise

Def. (Strictly) Characteristic Subgroup

Let H \leq G, then we say H is a (strictly) characteristic subgroup of G if H is invariant with respect to \text{Aut }G, that is H \leq_\text{Aut} G.

_Def. Fully Characteristic Subgroup

Let H \leq G, then we say H is a characteristic subgroup (or fully invariant subgroup) of G if H is invariant with respect to \text{End }G, that is H \leq_\text{End} G.

Thm. Characteristic Normality

Let H be a characteristic subgroup of G, then H is normal in the whole group, that is A \trianglelefteq N for all N \trianglelefteq G.

Exercise

Def. Complete Group

A group G is called complete if it has trivial center and \text{Aut }G = \text{Inn }G. Therefore,

Exercises

Def. Permutation

A permutation \sigma on a set X is a bijective function from X to X. The permutation x \mapsto x will be called the identity permutation.

We say an element x \in X is fixed under \sigma if \sigma(x) = x. Similarly, we say x is moved by \sigma if \sigma(x) \neq x.

For simplicity, we will use the set \textbf{I}_n = \Set{1, 2, ..., n} instead of any X of any cardinality.

More formally, we could make use of Well-Ordering Principle, initial segments, and ordinals. For now, this definition should suffice.

Def. Support

The support of a permutation \sigma denoted by \text{supp }\sigma is defined as the set of elements that are moved by \sigma, that is

Similarly, the set of fixed elements denoted with \text{fix }\sigma is the set

Def. Disjoint Permutations

The permutations \sigma_1, \sigma_2, ..., \sigma_n are said to be disjoint if their support is disjoint.

Def. Cycle

Let \tau be a permutation on \mathbf{I}_n with the support \Set{k_1, k_2, ..., k_r}. Then \tau is said to be a cycle (or cyclic) of length r if

denoted with (k_1 k_2 \cdots k_r).

A cycle of length r will be called a r-cycle. A 2-cycle is called a transposition.

There is no widespread consensus on how to explicitly define a cycle, but the intuition should be clear.

Def. Symmetric Group

Set of all permutations (bijections) on \textbf{I}_n will be denoted with \textbf{S}_n and it forms a group under function composition (exercise) called the symmetric group (of n letters).

Notice that \textbf{S}_n is of order n!.

Thm. Permutations are (Unique) Product of Disjoint Cycles

Every non-identity permutation in \textbf{S}_n is uniquely (up to the order of the factors) a product of disjoint cycles, each of which has length at least 2.

Def. Odd and Even

A permutation is said to be even (resp. odd) if it can be written as a product of even (resp. odd) number of transpositions.

Thm. Exclusively Odd or Even

A permutation \sigma \in \textbf{S}_n where n \geq 2 is either even or odd, but not both.

Therefore, the sign of a permutation \sigma denoted \text{sgn } \sigma is defined to be 1 if even and -1 if odd.

Thm. Alternating Group

Let \textbf{A}_n denote the set of all permutations of \textbf{S}_n. Then \textbf{A}_n is a normal subgroup of \textbf{S}_n of index 2. Moreover, \textbf{A}_n is the only subgroup of \textbf{S}_n of index 2.

\textbf{A}_n is called the alternating group (of degree n).

Thm. Galois

The alternating group \textbf{A}_n is simple if and only if n \neq 4.

Thm. Hölder

The symmetric group \mathbf{S}_n is complete if n \neq 2, 6.

Check out Kargapolov pp. 43-44 for the partial proof.

Thm. Dihedral Group Generators

Let n \geq 3, then the dihedral group \textbf{D}_n (which is of order is 2n) is a group whose generators a and b satisfy

1. a^n = b^2 = e and a^k \neq e if 0 < k < n,
2. aba = b.

Moreover, for n \geq 3, any group G which is generated by a and b that satisfy (1) and (2) is isomorphic to \textbf{D}_n.

Exercise. Generator of \textbf{D}_n

Let \Braket{a} \leq \textbf{D}_n for a \in \textbf{D}_n and |a|=n, then

1. \Braket{a} \trianglelefteq \textbf{D}_n, and
2. \textbf{D}_n/\Braket{a} = \Z_2.

Thm. Center of \textbf{D}_n

Let Z be the center of the group \textbf{D}_n, then

• Z = \Braket{e} if n is odd,
• Z \cong \Z_2 if n is even.

Def. Direct Product (of Groups)

This is equivalent to the formal definition of set of tuples from the axiomatic set theory, but for the family of groups instead of family of sets.

Let \Set{G_i} be a family of groups indexed by a non-empty set I, then the direct product (or complete direct sum) of the groups G_i denoted with \prod_{i\>\in\>I} G_i is the set of all functions

such that f(i) \in G_i. Notice that since each G_i is a group, thus non-empty, we have \prod G_i \neq \varnothing.

As a mental image, think of \prod G_i as the set of all (ordered) tuples where each i-th element belongs to G_i so that each f \in \prod G_i represent a tuple in that set.

Def. Natural Projections

Let \Set{G_i} be a non-empty family of groups, then \prod G_i is a group under component-wise multiplication and for each k \in I, the map

called the (natural) projection(s) of the direct product is an epimorphism of groups.

Exercise

Def. (External) Weak Direct Product

Let \Set{G_i} be a non-empty family of groups, then the (external) weak direct product of the groups G_i denoted with \prod^w G_i is the set of all f \in \prod G_i such that f(i) = e_i for all but a finite number of i \in I.

That is, non-identity elements of the tuple f are finite. Tuple consists of "mostly" identity elements.

Notice that if I is finite, then every direct product is a weak direct product.

Moreover, if each G_i is additive (that is abelian) \prod^w G_i is called the (external) direct sum denoted with \sum G_i.

Thm. Normals and Injections

Let \Set{G_i} be a family of non-empty groups, then

1. \prod^w G_i \trianglelefteq \prod G_i,
2. for each k \in I, the map \def\arraystretch{1.5} \begin{array}{lrll} i_k: & G_k & \to & \prod^w G_i \\ & a & \mapsto & f = (e_1, ..., e_{k-1}, a, e_{k+1}, ...) \end{array} is a monomorphism of groups,
3. for each k \in I, we have i_k(G_k)\trianglelefteq \prod G_i.

Exercise

Thm. Direct Sum and Family of Homomorphisms

Let \Set{A_i} be a non-empty family of abelian groups, and B an abelian group. If \Set{\varphi_i: A_i \to B} is a family of homomorphisms (with the same index set), then there exists an unique homomorphism

such that \varphi \circ i_k = \varphi_k for all k \in I. This property determines \sum A_i uniquely up to isomorphism.

This theorem is false if the groups are not abelian.

Thm. Direct Sum of Normals

Let \Set{N_i} be a non-empty family of normal subgroups of a group G such that

• G = \Braket{\>\bigcup N_i\>}, and
• for each k \in K, we have N_k \cap \Braket{\>\bigcup_{i\>\neq\>k} N_i\>} = \Braket{e}.

Then

and \Set{N_i} is called a normal decomposition of G.

Def. Internal Product

Let \Set{G_i} be a non-empty family of groups and \prod G_i = G. If \Set{G_i} is a normal decomposition of G, then G=\prod G_i is said to be the internal weak direct product (or internal direct sum if G is abelian).

Thm. Normal Decomposition Condition

Let \Set{N_i} be a non-empty family of normal subgroups of G. Then, \Set{N_i} is a normal decomposition of G if and only if for each non-identity g \in G is the unique product

where each i_k \in I is distinct and e \neq a_{i_k} \in N_{i_k} for each k = 1,2,...,n.

Exercise

Thm. Internal Direct Sum and Family of Homomorphisms

Let \Set{\varphi_i: G_i \to H_i} be a family of homomorphism of groups and let

Then \varphi is a homomorphism of groups such that

and

and

Moreover, \varphi is a monomorphism (resp. epimorphism) if each \varphi_i is.

Corollary. Normals and Quotients

Let \Set{G_i} be a non-empty family of groups and \Set{N_i} be a non-empty family of normal subgroups (of same index) such that N_i \trianglelefteq G_i for all i \in I. Then

1. \prod N_i \trianglelefteq \prod G_i and {(\prod G_i)}/{(\prod N_i)} \cong \prod (G_i/N_i),
2. \prod^w N_i \trianglelefteq \prod^w G_i and {(\prod^w G_i)}/{(\prod^w N_i)} \cong \prod^w (G_i/N_i)

Exercise, use First Isomorphism Theorem.

Def. Free Generator

We say S is a free generator if for each s \in S there exists a corresponding distinct s^{-1} \in S^{-1} called the inverse of s such that S and S^{-1} are disjoint and |S| = |S^{-1}|. Moreover, the identity \epsilon of S is an element such that \epsilon \not\in S \cup S^{-1} whose inverse is itself.

We could have defined the free generator S more formally with tuples and bijective functions, but the notation becomes very cumbersome very quickly. Intuition and the way to formalize it should be clear.

Def. Word

Let S be a free generator, then a word on S is a countably finite sequence (a_1, a_2, ...), indexed by \N^+, where

• a_i \in S \cup S^{-1} \cup \{\epsilon\} for each i \in \N^+, and
• for some n \in \N^+ we have a_k = \epsilon for all k \geq n.

The constant sequence (\epsilon, \epsilon, ...) is called the empty word and denoted with 1.

A word w = (a_1, a_2, ...) is said to be reduced if

1. a_i = x \implies a_{i+1} \neq x^{-1} for all i \in \N^+ and x \in S \cup S^{-1}, that is there are no adjacent inverses (other than \epsilon),
2. a_k = \epsilon \implies a_i = \epsilon for all i \geq k, that is any identity is followed by an identity.

In particular, every non-empty reduced word is of the form, for some n \in \N^+

where x_i \in X and \lambda_i = \pm 1. From now on we will omit the parentheses.

Notice that the empty word is reduced and x^1 := x. Reduction algorithm should be obvious.

This is a rather formal definition of a word, simply put a word on X is a finite product of elements X \cup X^{-1} such that inverses cancel each other out where \epsilon is the identity. Now, we should define the binary multiplication on (reduced) words themselves to make it a group.

Def. Free Group

Let non-empty X be a free generator, and let \mathbf{F}(X) be the set of all reduced words on X, then \mathbf{F}(X) is a group under the binary operation where xy is the reduced concatenation for all x,y \in \mathbf{F}(X). The group \mathbf{F}(X) is called the (absolutely) free group on the set X denoted by \Braket{X}. The cardinal number |X| is called the rank of the free group \mathbf{F}(X).

For a more formal definition check out Hungerford pp. 64-65

Do not mistake \Braket{\cdot} here with the notation of cyclic groups or generators.

Notice that the representation \Braket{X \mid R} is isomorphic to \mathbf{F}(X) / \Braket{\Braket{R}}.

One usually denotes a free group of rank n with \mathbf{F}_n or \mathbf{F}_\infty if X is (countably) infinite.

Thm. Universal Property

Let X be a set, \Braket{X} the free group generated by X, and i: X \to \Braket{X} an inclusion map. For G a group and \varphi: X \to G a map of sets (a map without any extra structure), there exists an unique homomorphism of groups \bar{\varphi}: \Braket{X} \to G such that \bar{\varphi} \circ i = \varphi.

Def. Presentation

Let Y \subseteq \Braket{X}, then a group G is said to be defined by the generators x \in X and relations w \in Y provided that N \trianglelefteq \Braket{X} is generated by Y. Noting that G \cong \Braket{X}/N, we say \Braket{X | Y} is a presentation of G.

Moreover, instead of \Braket{X|Y}, we may write \Braket{X|w_1=1, w_2 = 1, ...} for brevity, or even more compactly \Braket{X|w_1, w_2, ...}

We have previously shown such defined group exists and it is the largest possible group in that sense.

Thm. Van Dyck

Let G = \Braket{X|Y} and H = \Braket{X} such that H satisfies all the relations w = 1 where w \in Y, then there is an epimorphism \psi: G \to H.

Thm. Nielsen-Schreier

Proof of subgroups of free groups are also free was provided for finitely generated free group by J. Nielsen in 1921 and for all any free group by O. Schreier in 1927.

Let G be a free group of rank r and H \leq G, then H is itself isomorhpic to a free group. Moreover, if H is of finite index m, then H is of rank 1 + m(r - 1).

Thm. Free Groups are Residually Finite p-groups

Every free group F(X) is residually a finite finite p-group where p is prime.

Def. Free Product

TODO:

Exercises

Def. Basis

Noting that the subgroup \Braket{X} generated by X in additive notation consists of all linear combinations

where n_i \in \Z and x_i \in X.

A basis of an abelian group F is a subset X of F such that

• F = \Braket{X}, and
• for distinct x_1, ..., x_n \in X and n_i \in \Z we have

Def. Free Abelian Group

Let F be an abelian group, then it is called a free abelian group if it has an non-empty basis (or equivalently any one of the conditions below).

By definition, the trivial group \Braket{0} is the free abelian group on the null set \varnothing.

Thm. Equivalent Basis Conditions

Let F be an abelian group, then the following are equivalent

1. F has a non-empty basis,
2. F is the (internal) direct sum of a family of infinite cyclic subgroups,
3. F is (isomorphic to) a direct sum of copies of (\Z, +),

Thm. Basis Cardinality

Any two bases of a free abelian group F have the same cardinality called the rank of F.

Thm. Free Abelian Subgroups

Every subgroup of a free abelian group including the \Braket{0} is also free abelian.

Thm. Isomorphism on Free Abelian Groups

Two free abelian groups are isomorphic if and only if they have the same rank.

Thm. Free Abelian Groups and Abelian Groups

Every abelian group G is the homomorphic image of a free abelian group of rank |X| where X is a set of generators of G.

Thm. Basis for Subgroups

Let F be a free abelian group of finite rank n with the basis \Set{x_1, ..., x_n} and G its non-zero subgroup, then there exists an integer r \leq n and positive integers d_1,...,d_r such that d_1\>|\>d_2\>|\>\cdots\>|\>d_r where G is free abelian with the basis \Set{d_1x_1, ..., d_rx_r}.

Def. General Abelian Basis

We have defined basis above for free abelian groups which doesn't really make sense for any abelian group, therefore...

Let G be an abelian group. We say a finite set of elements g_1,..., g_k \in G is linearly dependent (over \Z) if there exists n_1, ..., n_k \in \Z not all zero such that

Moreover, an arbitrary family of elements of G is said the be linearly dependent if some finite subfamily is linearly dependent.

For any abelian group A, not necessarily torsion-free, which contain at least one element of infinite order there is always a maximal linearly independent subset. Moreover, all such maximal subsets have the same cardinality \kappa. In this case, we say A is of rank \kappa.

Since periodic (every element has finite order) abelian groups do not contain any linearly independent subset we define their ranks to be 0.

Thm. A Rank 1 Abelian Group

A nonzero torsion-free abelian group has rank 1 if and only if it is isomorphic to a subgroup of (\mathbb{Q}, +).

Thm. Cyclic Decomposition

Let G be a finitely generated abelian group, then

where each C_i is a cyclic group with finite ones of order m_i > 1 to m_k such that

Thm. Cyclic Decomposition 2

Let G be a finitely generated abelian group, then

where each C_i is either infinite or of order a power of a prime.

Notice that this theorem implies if positive integer m divides the order of a group, then there exists a subgroup of order m. This is not the case for any group, we have just proved this for finite abelian groups. We'll touch this subject later on Sylow Theorems.

Thm. Prime Decomposition and Cyclics

Let m = p_1^{n_1} p_2^{n_2} \cdots p_k^{n_k} be a positive integer such that each p_i is a distinct prime and n_i > 0, then

Corollary. FGA Subgroups

Let G be a finitely generated abelian group of order n, then G has a subgroup of order m for every positive integer m that divides n.

Def. Divisible Group

A group G is said to be divisible (or radicable) if for every n \in \Z^+ and every element g \in G the equation

has at least one solution in G.

Thm. Divisible Embedding of Abelian Groups

Every abelian group can be embedded in some divisible abelian group.

Thm. Divisible Subgroups

A divisible subgroup D of an abelian group G is a direct summand of G, that is G = D \oplus B for some subgroup B of G.

Existence of such B follows from Zorn's Lemma.

Thm. Divisible Group Decomposition

Every divisible group G can be decomposed as a direct sum of subgroups each isomorphic to either (\mathbb{Q}, +) or the quasicyclic (Prüfer) group \mathbf{C}_{p^\infty} where p may vary for different summands.

Exercises

Def. Torsion Subgroup

Let G be an abelian group and T the set of all elements of finite order, then T is a subgroup of G called the torsion subgroup of G. Moreover, G/T is also torsion-free.

Note that, in general, the torsion subgroup is not a direct factor.

Thm. Prüfer's First Theorem

First check out Exercise 1 below.

An abelian p-group of finite exponent is a direct sum of cyclic subgroups.

Def. Pure Subgroup

A subgroup A of a group G is said to be pure if for every n \in \Z and a \in A, the equation

is soluble in G implies it is soluble in A.

Thm. L. Ja. Kulikov-Prüfer

You may want to solve exercises first.

If A is a pure subgroup of finite exponent of an abelian group G, then A is a direct summand of G.

By exercise 3

In particular, the torsion subgroup T of abelian group G is a direct summand of G.

Def. Height

An non-identity element g of an abelian p-group G is of finite height h in G if the equation

has a solution only for n \leq h. Moreover, the height of g is defined to infinite if it has a solution for every n.

Thm. Prüfer's Second Theorem

A countable abelian p-group without elements of infinite height is a direct sum of cyclic subgroups.

Exercises

Def. Indecomposable

A group G is called indecomposable if G \neq \Braket{e} and G is not the (internal) direct product of two of its proper subgroups.

Def. Chain Conditions

Let G be a group, then we say

• G satisfies ascending chain condition (ACC) on subgroups if every chain G_1 \leq G_2 \leq \cdots of subgroups of G there exists n such that G_i = G_n for all i \geq n.
• G satisfies descending chain condition (DCC) on subgroups if every chain G_1 \geq G_2 \geq \cdots of subgroups of G there exists n such that G_i = G_n for all i \geq n.

Thm. Normal Chain Condition and Indecomposition

Let G be a group which satisfies ACC or DCC on normal subgroup, then G is a direct product of a finite number of indecomposable subgroups.

Def. Normal Endomorphism

Let \varphi: be an endomorphism of G. We say \varphi is a normal endomorphism if

that is

for all a,b \in G.

Thm. Chain Conditions and Morphisms

Let G be a group that satisfies ascending (resp. descending) chain condition on normal subgroups and \varphi a normal endomorphism of G. Then \varphi is an automorphism of G if and only if \varphi is an epimorphism (resp. monomorphism).

Thm. Fitting

Let G be a group that satisfies both ACC and DCC on normal subgroups and \varphi a normal endomorphism of G, then for some n \geq 1

Def. Nilpotent Endomorphism

Let \varphi be an endomorphism of G, then it is said to be nilpotent if there exists n > 0 such that

for all x \in G.

Thm. Krull-Schmidt

Let G be a group that satisfies both ACC and DCC on normal subgroups. Moreover, let each G_i and H_j be indecomposable where

then s = t and for some reindexing G_i \cong H_i.

Def. Group Action

Let G be a group and X any set. A binary operation *: G \times X \to X is called a (left) action if, for all a,b \in G and x \in X:

1. a * (b * x) = (a b) * x, and
2. e * x = x

where (1) is called identity property and (2) is called compatibility property.

When such action is given, we say G acts on the set X.

For establishing general properties of group actions, it suffices to consider only left actions.

Def. Orbits

Let the group G act on a set X, then the orbit of an element x \in X is the set of elements

The orbit of x can be denoted with \bar{x} since, as shown below, orbits partition G.

Def. Transitivity

The group action is said to be transitive if for x, y \in X there exists g \in G so that g * x = y.

Def. Stabilizer

Let G act on X and x \in X, then the stabilizer subgroup of G with respect to x is defined as

Thm. Basic Orbit and Stabilizer Properties

Let the group G act on a set X and x \in X, then

1. Set of orbits partition the set X.
2. The group action is transitive if and only if it has exactly one orbit.
3. If the action is transitive, then there is exactly one orbit, so that G * x = G for all x \in X.
4. G_x \leq G.

Since G_x is a subgroup of G, stabilizer is also called the subgroup fixing x or the isotropy group of x.

Thm. Orbit-Stabilizer Theorem

Let G be a finite group that acts on a set X and x \in X, then

Corollary. Normalizer, Centralizer, and Conjugacy Classes

Let G be a finite group and K \leq G, then

1. |\bar{x}| = |G:C_G(x)| and |\bar{x}| divides |G|,
2. Let \bar{x_1}, \bar{x_2}, ..., \bar{x_n} be all the distinct conjugacy classes of G, then

3. The number of subgroups of G conjugate to K is |G : N_G(K)|, which divies |G|.

Thm. Action Induced Homomorphism

Let G act on a set X, then this action induces a homomorphism \varphi: G \to \mathbf{P}(X), where \mathbf{P}(X) is the group of all permutations of X.

Thm. Cayley

Let G be a group, then there exists a monomorphism \varphi: G \to \mathbf{P}(G). Therefore, every group is isomorphic to a group of permutations. In particular, every finite group of order n is isomorphic to a subgroup of \mathbf{S}_n.

Corollary. Introduction to Inner Automorphism

Let G be a group, then

1. For each h \in G, conjugation by h induces an automorphism of G.
2. There exists a homomorphism G \to \text{Aut }G whose kernel is C(G).

We will see this in more detail in the next sections.

Def. Translation

Let G be a group and H \leq G. The action of H on the set G which is given by (h,g) \mapsto hg, where hg is the group product is called a (left) translation.

Let S be the set of all (left) cosets of H in G, then we say H acts on S by translation so that (h, gH) \mapsto hg H.

Def. Invariant

Let G be a group that acts on a set X and g \in G, then the (left) invariant by g denoted by X^g is the set of elements in X that are fixed by g, that is

Do not mistake this with the stabilizer subgroup.

Thm. Burnside's Lemma

Let G be a finite group acting on a set X, then

where |X/G| denotes the number of orbits.

Lemma 1

Let H be a group of order p^n that acts on finite set S where p is prime, if

then |S| \equiv_p |S_0|.

Thm. Cauchy

Let G be a group such that for some prime p divides |G|, then G has an element of order p.

Def. p-group

Let p be any prime. A group in which every element is with order p^k where k \geq 0 is called a p-group. If a subgroup is a p-group of a group, then it is called a p-subgroup.

Notice that \Braket{e} is a p-subgroup of any group for any prime p.

Thm. p-groups

A finite group G is a p-group if and only if order of G is a power of p.

Lemma 2

If H is a p-subgroup of a finite group G, then

Moreover if p divides |G:H|, then

Thm. First Sylow Theorem

Let p be a prime and n \geq 1. If G be a group of order p^n m where (p,m)=1, then

1. G contains a subgroup of order p^i for each 1 \leq i \leq n, and
2. Every subgroup of G of order p^i is normal in some subgroup of order p^{i+1}.

(1) is also called the existence and (2) is called the inclusion theorem of Sylow.

Def. Sylow p-subgroup

A subgroup P of a group G is said to be a Sylow p-subgroup of G if P is a maximal p-subgroup.

Thm. Basic Sylow p-subgroup Properties

Let G be a group of order p^n m where (p, m) = 1 and H is its p-subgroup, then

1. H is a Sylow p-subgroup if and only if |H|=p^n,
2. Every conjugate of a Sylow p-subgroup is a Sylow p-subgroup,
3. If there is only one Sylow p-subgroup P, then P \trianglelefteq G.

Thm. Second Sylow Thorem

If H is a p-subgroup of a finite group G, and P is any Sylow p-subgroup of G, then there exists x \in G such that

In particular, any two Sylow p-subgroups of G are conjugate.

This is also called the Sylow's conjugacy theorem.

Thm. Third Sylow Theorem

Let G be a finite group, then the number of Sylow p-subgroups of G divides |G| and is of the form kp + 1 for some k \geq 0.

This can also be called the Sylow's number (or counting) theorem.

Thm. Idempotency of Normalizer on Sylow p-subgroups

If P is a Sylow p-subgroup of a finite group G, then

Exercises

Def. \mathbf{PSL}_n(K)

By Jordan (1870)

Let K be a field, then the quotient group of the special linear group \mathbf{SL}_n(K) by it's center (the scalar matrices) is called the projective special linear group of degree n over K denoted by \mathbf{PSL}_2(K).

Thm. Jordan-Dickson

Let K be any field, then \mathbf{PSL}_n(K) is simple with exceptions \mathbf{PSL}_2(2) and \mathbf{PSL}_2(3).

Exercises

Def. Commutator

Let G be a group and a, b \in G. Obviously, two elements a and b commute if and only if a^{-1}b^{-1}ab = e. The left-hand side of this equation will be denoted with [a,b] called the commutator of a and b, that is

For A, B \subseteq G, we define mutual commutator subgroup as

More generally,

and

Thm. Basic Commutator Properties

Let G be a group and a,b,c, x \in G, then

• [a,b] = e if and only if ab=ba, indeed
• e is the only commutator if and only if G is abelian,
• [a,b]^{-1} = [b,a],
• [a,b]^x = [a^x, b^x],
• [ab,c]=[ac]^b[b,c],
• [a^{-1}, b] = [b,a]^{a^{-1}},
• For any group homomorphism \phi: G \to H, we have \phi([a,b]) = [\phi(a), \phi(b)].

The product of two or more commutators need not be a commutator. Indeed, it is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property — See Stack Exchange: Mariano Suárez-Álvarez.

Def. Commutator Subgroup and Derived Series

Let G be a group. Then the commutator subgroup (or derived subgroup) of G denoted with G' or G^{(1)} is the (normal) subgroup [G, G].

Applied recursively, we get the derived series of the group G

where i-th derived subgroup G^{(n)} is defined as

For a finite group this series terminates, to what is called a perfect group which may be trivial or not. Moreover, realize that G is abelian if and only if G' = \Braket{e}. In a sense, G' provides a measure of how much G differs from an abelian group.

Thm. Some Commutator Subgroup Properties

Let G be a group and N \trianglelefteq G, then

1. G' \trianglelefteq G,
2. G/G' is abelian,
3. G/N is abelian if and only if G' \subseteq N.

Thm. Three Commutator Lemma

Let G be a group, A, B, C \leq G, and N \trianglelefteq G. If any two commutator subgroups

lie in N, then so is the other one.

Exercises

Def. Central Series

Let G be a group, a central series of G is a sequence of subgroups such that

where

for each i.

Following definitions and their naming will make more sense later on.

The lower (or descending) central series of G is a sequence of subgroups

where

Do not confuse this with the derived series we have defined as G^{(n)} := [G^{(n-1)}, G^{(n-1)}] earlier.

Similarly, upper (or ascending) central series of G is the sequence

where, similarly

Exercise that indeed each subgroup is indeed a normal subgroup.

Thm. TFAE: Nilpotent Groups

A group G is called nilpotent if it satisfies any of equivalent definitions

1. G has a central series of finite lenght.
2. G has a lower central series finitely terminating in the trivial subgroup, that is D_n = \Braket{e} for some n.
3. G has an upper central series finitely terminating in the whole group, that is A_n = G for some n.

Thm. Finite p-groups are Nilpotent

Every finite p-group is nilpotent.

Thm. Finite Direct Product of Nilpotents are Nilpotent

The direct product of a finite number of nilpotent groups is nilpotent.

Thm. Nilpotency and Sylow Subgroups

A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.

Def. Solvable Groups

A group G is said to be solvable if G^{(n)} = \Braket{e} for some n.

Thm. Nilpotent \implies Solvable

Every nilpotent group is solvable.

Thm. Solvable Group Properties

1. Let G be a solvable group, then every subgroup and every homomorphic image of G is also solvable.
2. Let N \trianglelefteq G such that N and G/N are solvable, then G is solvable.

Thm. Solvability of \mathbf{S}_n

The symmetric group \mathbf{S}_n is not solvable if n \geq 5.

Thm. More on Solvable Groups

Let N be a normal subgroup of a finite group G and H \leq G, then

1. If H is a characteristic subgroup of N, then H \trianglelefteq G,
2. Every normal Sylow p-subgroup of G is fully invariant,
3. If G is solvable and N is a minimal normal subgroup, then N is an abelian p-group for some prime p.

Thm. (P. Hall)

Let G be a finite solvable group of order mn such that (m,n)=1, then

1. G contains a subgroup of order m,
2. Any two subgroups of G of order m are conjugate,
3. Any subgroup of G of order k such that k \mid m is contained in a subgroup of order m.

This theorem is difficult to prove, so if you are interested check out Hungerford pp. 104-106.

Def. Normal Series

Let G be a group. A subnormal series (or just series) is a sequence of subgroups such that each N_{i+1} is normal in N_{i}, that is

The factors (or factor groups) of the series are the quotient groups N_{i+1}/N_{i}. The length of the series is the number of strict inclusions (or equivalently, the number of non-identity factors).

Generally, n could be any ordinal.

Def. Composition and Solvable Series

A subnormal series (G)_i is called composition series if each factor G_i/G_{i+1} is simple.

Moreover, (G)_i is called solvable series if each factor is abelian.

Thm. Solvability

A group G is solvable if and only if it has a solvable series.

Thm. Jordan-Hölder

Any two composition series of a group G are equivalent.

Note that this theorem does not assert the existence of composition series.

Def. (Outer) Semidirect Product

Let G and H be groups and \theta: H \to \text{Aut }G a homomorphism. Let G \rtimes_\theta H be the set G \times H with the binary operation

So that G \rtimes_\theta H is group with the identity (e_G,e_H) and

G \rtimes_\theta H is called the (outer) semidirect product of G and H with respect to \theta.

Thm. Normal Complement

Let N \trianglelefteq G, then the following are equivalent

1. G = NH and N \cap H = \{e\} for some H \leq G.
2. For each g \in G, there are unique n \in N and h \in H such that g=nh.

Def. Inner Semidirect Product

Let G be a group such that N \trianglelefteq G and H \leq G are complements in G, then define

The semidirect product N \rtimes_{\varphi} H denoted simply by N \rtimes H or H \ltimes N is called the inner semidirect product of N and H, so that G = N \rtimes H. We also say G is a semidirect product of H acting on N.

Def. Holomorph

Let G be a group, then the holomorph of G is defined as

whose multiplication simplifies to

Notation. Cartesian and Direct Product

Let I be an index set, then

• A^{[I]} denotes the |I|-fold cartesian product, and
• A^{(I)} denotes the |I|-fold direct product.

Def. Wreath Products

Let G and H be groups such that H acts on \Omega from left.

We can extend the action of H on \Omega to an action on G^{[\Omega]} via

for all h \in H and all (g_w)_{w \in \Omega} \in G^{[\Omega]}.

The unrestricted wreath product is defined as

and the subgroup G^{[\Omega]} of G^{[\Omega]} \rtimes H is called the base of the wreath product.

Similarly, the restricted wreath product denoted with \text{wr}_{\Omega} is the product defined above with G^{(\Omega)} instead of G^{[\Omega]}.

Two definitions coincide when \Omega is finite.

If \Omega is not explicitly stated, we take \Omega = H.

Either variant is denoted with \wr_\Omega.

Thm. Wreath Properties

Let G and H be groups, and H acts on \Omega, then

1. G \enspace \text{wr}_\Omega \enspace H \leq G \enspace \text{Wr}_\Omega \enspace H
2. |G \wr_\Omega H| = {|G|}^{|\Omega|} |H|

Thm. Kaluznin-Krasner

Every extension of a group G by a group H can be embedded in the unrestricted wreath product G \enspace \text{Wr} \enspace H.

Def. Residually Finite

We say a group G is residually finite if for each 1 \neq g \in G there exists N \trianglelefteq G of finite index such that g \notin N.

Thm. Equivalent Residually Finite Conditions

A group G is residually finite if and only if either

• for every 1_G \neq g \in G there is a group homomorphism \varphi: G \to H where H is a finite group and \varphi(g) \neq 1_H.
• the intersection of all its subgroups of finite index is trivial.
• the intersection of all its normal subgroups of finite index is trivial.
• G can be embedded inside the direct product of a family of finite groups.

Def. Perfect Group

Thm. Dedekind Modular Law (Identity)

See https://math.stackexchange.com/questions/3957388/intuition-behind-dedekinds-modular-law

Exercises