Groups

Overview

A group $⟨ G, * ⟩$ is a monoid with inverse elements. That is, for each $a \in G$ , there is an element $a^{'} \in G$ such that $a * a^{'} = a^{'} * a = e$ . Oftentimes we simply write "a group $G$ " with the understanding there is some relevant binary operation on set $G$ .

Addition is usually denoted using the $+$ symbol. This does not necessarily correspond to the standard additive operation. It is generally reserved for commutative operations. An inverse of $a$ in additive notation is denoted as $- a$ . The identity element is denoted as $0$ . For some $n \in Z$ ,

$0 a$ is interpreted as the identity element.
$n a$ is interpreted as $a + a + \dots + a$ , where $a$ is repeated $n$ times.
$- n a$ is interpreted as $(- a) + (- a) + \dots + (- a)$ , where $(- a)$ is repeated $n$ times.

Multiplication is usually denoted using the $\cdot$ symbol or the immediate juxtaposition of two elements. This does not necessarily correspond to the standard multiplicative operation. An inverse of $a$ in multiplicative notation is typically denoted as $a^{- 1}$ . An identity element is typically denoted as $1$ . For some $n \in Z$ :

$a^{0}$ is interpreted as the identity element.
$a^{n}$ is interpreted as $(a) (a) \dots (a)$ , where $a$ is repeated $n$ times.
$a^{- n}$ is interpreted as $(a^{- 1}) (a^{- 1}) \dots (a^{- 1})$ , where $a^{- 1}$ is repeated $n$ times.

A group is abelian if its binary operation is commutative. Otherwise it is nonabelian.

Cancellation

Let $⟨ G, * ⟩$ be a group and $a, b, c \in G$ . The left and right cancellation laws respectively state that

\begin{array}{r} a * b = a * c ⟹ b = c \\ b * a = c * a ⟹ b = c \end{array}

These implications prove uniqueness of linear equations.

Order

A group's order refers to the cardinality of its associated set. If the associated set is not finite, the group is said to have infinite order. Up to isomorphism:

Order	Groups
$0$	-
$1$	${e}$ , the trivial group
$2$	$Z_{2}$
$3$	$Z_{3}$
$4$	$Z_{4}$ and $K_{4}$ (or $V$ ), the Klein group
$\dots$	$\dots$

By Lagrange's theorem, every group of prime order is cyclic. Hence, there is only one group structure, up to isomorphism, of a given prime order $p$ .

Cayley Tables

A Cayley table describes the structure of a finite group by arranging all possible products of all the group's elements in a square table.

A group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. No row nor column can contain the same element twice.

Subgroups

Let $⟨ G, * ⟩$ be a group. If $H \subseteq G$ is a group under $*$ , then $H$ is a subgroup of $G$ .

An improper subgroup of $G$ is $G$ itself. Every other subgroup is a proper subgroup. The subgroup ${e}$ , where $e$ denotes the identity element, is the trivial subgroup. Every other subgroup is nontrivial.

Criterion

A nonempty subset $H$ of a group $G$ is a subgroup if and only if

$H$ is closed under multiplication, and
$H$ is closed under inverses, i.e. $\forall a \in H, a^{- 1} \in H$ .

Product Sets

Let $A_{1}, A_{2}, \dots, A_{n}$ be subsets of a group $G$ . Then their product set is defined as

A_{1} \dots A_{n} = {a_{1} \dots a_{n} ∣ a_{1} \in A_{1}, \dots, a_{n} \in A_{n}} .

Sumsets

Let $A_{1}, A_{2}, \dots, A_{n}$ be subsets of an abelian group $G$ . Then their sumset (or Minkowski sum) is defined as

A_{1} + \dots + A_{n} = {a_{1} + \dots + a_{n} ∣ a_{1} \in A_{1}, \dots, a_{n} \in A_{n}} .

Unions

Let $G$ be a group and $H, K \leq G$ . Then $H \cup K$ is a subgroup of $G$ if and only if either $H \subseteq K$ or $K \subseteq H$ .

This does not generalize to more than two subgroups. For instance, $K_{4}$ is a union of three of its subgroups.

Intersections

Let ${H_{i} ∣ i \in I}$ be an indexed set of subgroups of $G$ . Then their intersection $⋂_{i \in I} H_{i}$ is a subgroup of $G$ .

Direct Products

Let $G_{1}, G_{2}, \dots, G_{n}$ be groups. Let $a_{i}, b_{i} \in G_{i}$ for $1 \leq i \leq n$ . Then direct product $⨂_{i = 1}^{n} G_{i}$ is a group under binary operation

(a_{1}, a_{2}, \dots, a_{n}) (b_{1}, b_{2}, \dots, b_{n}) = (a_{1} b_{1}, a_{2} b_{2}, \dots, a_{n} b_{n}) .

The foregoing definition is referred to as an external direct product. In contrast, a group $G$ is an internal direct product of normal subgroups $H_{1}, H_{2}, \dots, H_{n} ⊴ G$ if every $g \in G$ is uniquely expressed as $g = h_{1} h_{2} \dots h_{n}$ where $h_{i} \in H_{i}$ .

Internal Criterion

A group $G$ is an (internal) direct product of normal subgroups $H_{1}, H_{2}, \dots, H_{n} ⊴ G$ if and only if each of the following conditions hold:

$G = H_{1} H_{2} \dots H_{n}$ ;
$H_{i} \cap (H_{1} \dots H_{i - 1} H_{i + 1} \dots H_{n}) = {e}$ for $1 \leq i \leq n$ ;
$h_{i} h_{j} = h_{j} h_{i}$ for each $h_{i} \in H_{i}$ , $h_{j} \in H_{j}$ , $1 \leq i < j \leq n$ .

Furthermore, product set $\prod H_{i}$ is an (internal) direct product if and only if $e \in \prod H_{i}$ is uniquely written as $(e) (e) \dots (e)$ .

Decompositions

A group $G$ is decomposable if it is isomorphic to a direct product of two proper, nontrivial subgroups. Otherwise $G$ is indecomposable.

The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime. This follows by the fundamental theorem of finitely generated abelian groups and the fact every cyclic group of order $n$ is isomorphic to $⟨ Z_{n}, +_{n} ⟩$ .

Homomorphisms

Let $G$ and $G^{'}$ be groups. A group homomorphism $ϕ : G \to G^{'}$ is a mapping that satisfies the homomorphism property:

ϕ (a * b) = ϕ (a) *^{'} ϕ (b) .

Then the following structural properties are preserved:

Identity
- If $e \in G$ is identity, then $ϕ (e) \in G^{'}$ is identity.
Inverses
- If $a \in G$ , then $ϕ (a^{- 1}) = ϕ (a)^{- 1}$ .
Subgroups
- If $H \leq G$ , then $ϕ [[H]] \leq G^{'}$ .
- If $H^{'} \leq G^{'}$ , then $ϕ^{- 1} [[H^{'}]] \leq G$ .
- If $N ⊴ G$ , then $ϕ [[N]] ⊴ ϕ [[G]]$ .
- If $N^{'} ⊴ ϕ [[G]]$ , then $ϕ^{- 1} [[N^{'}]] ⊴ G$ .

Images

Let $ϕ : G \to G^{'}$ be a group homomorphism. The image of $ϕ$ is defined as

Im (ϕ) = ϕ [[G]] = {ϕ (g) ∣ g \in G} .

If $H = Ker (ϕ)$ , then $μ : G / H \to ϕ [[G]]$ defined as $μ (a H) = ϕ (a)$ is an isomorphism. This particular isomorphism is called the canonical isomorphism of $ϕ$ .

Kernels

Let $ϕ : G \to G^{'}$ be a group homomorphism. The kernel of $ϕ$ is defined as

Ker (ϕ) = ϕ^{- 1} [[{e^{'}}]] = {g \in G ∣ ϕ (g) = e^{'}} .

The fiber of $ϕ (a)$ is the left coset $a H$ of $H$ , where $H = Ker (ϕ)$ . It also coincides with the right coset $H a$ of $H$ . That is, $ϕ^{- 1} [[{ϕ (a)}]] = a H = H a$ . Hence the coimage of $ϕ$ corresponds to the left (or right) cosets of $H$ .

A group homomorphism $ϕ : G \to G$ is injective if and only if $Ker (ϕ) = {e}$ .

Fundamental Homomorphism Theorem

Let $ϕ : G \to G^{'}$ be a group homomorphism with kernel $H$ . Let $μ : G / H \to ϕ [[G]]$ be its canonical isomorphism and $γ : G \to G / H$ be its canonical homomorphism. Then $ϕ (g) = μ γ (g)$ for each $g \in G$ .

Homomorphisms Group

Let $G$ be a group and $H$ be an abelian group. Then homomorphism set $Hom (G, H)$ is an abelian group. Compare this to the analogous result in linear algebra.

Cosets

Let $H$ be a subgroup of a group $G$ . The left coset of $H$ containing $a$ is the subset $a H = {a h ∣ h \in H}$ . The right coset of $H$ containing $a$ is the subset $H a = {h a ∣ h \in H}$ .

These sets correspond to the cells of the following equivalence relations:

$a \sim_{L} b$ if and only if $a^{- 1} b \in H$ ;
$a \sim_{R} b$ if and only if $a b^{- 1} \in H$ .

The set of left cosets of $H$ in $G$ is typically denoted as $G / H$ . The set of right cosets of $H$ in $G$ is typically denoted as $H ∖ G$ .

Lagrange's Theorem

Let $H$ be a subgroup of a finite group $G$ . Then the order of $H$ is a divisor of the order of $G$ .

Indices

Let $H$ be a subgroup of a group $G$ . The number of left cosets of $H$ in $G$ is the index $(G : H)$ of $H$ in $G$ .

Let $H$ and $K$ are subgroups of a group $G$ such that $K \leq H \leq G$ . If $(H : K)$ and $(G : H)$ are both finite, then $(G : K) = (G : H) (H : K)$ .