Group Theory

Some preliminaries

Set theory

A set is a collection of mathematical objects called its elements. We write $a \in X$ if $a$ is an element of the set $X$ and $a \notin X$ otherwise. A set is completely determined by its elements (axiom of extensionality), that is, if $X$ and $Y$ are sets then $$\forall a\ (a \in X \iff a \in Y) \implies X = Y.$$ The converse of this axiom is also true and defines the notion of equality.

We say that a set $B$ is a subset of another set $A$ and write $B \subseteq A$ if and only if $$a \in B \implies a \in A.$$ Set equality can thus be restated as $$A = B \iff A \subseteq B \text{ and } B \subseteq A.$$ This fact is a useful tool in proving set equalities.

The union of two sets $A$ and $B$ is the set $A \cup B = \{x : x \in A \text{ or } x \in B\}$, while the intersection of $A$ and $B$ is the set $A \cap B = \{x : x \in A \text{ and } x \in B\}$. We can extend this definition for an arbitrary collection $\mathfrak{S}$ of sets as follows: $$x \in \bigcup_{S \in \mathfrak{S}} S \iff \exists S \in \mathfrak{S}\ (x \in S)$$ and $$x \in \bigcap_{S \in \mathfrak{S}} S \iff \forall S \in \mathfrak{S}\ (x \in S).$$ Finally, the set difference of $A$ and $B$ is the set $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$.

An ordered pair $(a, b)$ is a set $\{\{a\}, \{a, b\}\}$. We can extend this definition to ordered $n$-tuples $(a_1, a_2, \ldots, a_n)$, which we define to be the set $\{\{a_1\}, \{a_1, a_2\}, \ldots, \{a_1, a_2, \ldots, a_n\}\}$. We say that two ordered $n$-tuples $(a_1, a_2, \ldots, a_n)$ and $(b_1, b_2, \ldots, b_n)$ are equal if and only if $a_i = b_i$ for all $1 \leq i \leq n$.

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$. We denote this set by $A \times B$.

A relation on a set $X$ is a subset $R$ of $X \times X$. If $(a, b) \in R$, we say that $a$ is related to $b$ by $R$ and write $R(a, b)$ or $aRb$. If $R$ is a relation on $X$, then we say that $R$ is reflexive if $aRa$ for all $a \in X$. We say that $R$ is symmetric if $aRb$ implies $bRa$ for all $a, b \in X$. We say that $R$ is transitive if $aRb$ and $bRc$ implies $aRc$ for all $a, b, c \in X$. If $R$ is reflexive, symmetric, and transitive, then we say that $R$ is an equivalence relation and we write $a \sim b$ instead of $aRb$. Given a set $X$ and an equivalence relation $\sim$ on $X$, the equivalence class of an element $a$ of $X$ (written $[x]$) is the set $$[a] = \{x \in X : a \sim x\}.$$

Mappings

A mapping (or a function) $f$ from a set $A$ to a set $B$ written $f : A \to B$ is a rule that assigns to each element $a \in A$ a unique element $b \in B$. We would often write $f(a)$ instead of $b$ and say that $f$ maps $a$ to $b$ (or in symbols $a \mapsto b$ or $a \mapsto f(a)$). The set $A$ is called the domain of $f$ and the set $B$ is called the codomain of $f$. The image of $f$ is the set $$f(A) = \{f(a) : a \in A\}.$$ If $X \subseteq A$, then the image of $X$ under $f$ is the set $$f(X) = \{f(x) : x \in X\}.$$ If $Y \subseteq B$, then the inverse image of $Y$ under $f$ is the set $$f^{-1}(Y) = \{x \in A : f(x) \in Y\}.$$ The restriction of $f$ to $X$ is the mapping $f|_X : X \to B$ defined by $f|_X(x) = f(x)$ for all $x \in X$.

A mapping $f : A \to B$ is injective if $f(a) = f(b)$ implies $a = b$ for all $a, b \in A$. A mapping $f : A \to B$ is surjective if $f(A) = B$. A mapping $f : A \to B$ is bijective if it is both injective and surjective. If $f : A \to B$ is bijective, then there exists a mapping $g : B \to A$ such that $g \circ f = \operatorname{id}_A$ and $f \circ g = \operatorname{id}_B$, where $\operatorname{id}_A$ and $\operatorname{id}_B$ are the identity mappings on $A$ and $B$, respectively. The mapping $g$ is called the inverse of $f$ and is denoted by $f^{-1}$.

1. The definition of a group

Observe that commutativity is not required for a group. That is, we do not require that $\varphi(a, b) = \varphi(b, a)$ for all $a, b \in G$. If $\varphi$ does, however, satisfy this property, then we say that the underlying set $G$ is an abelian group under $\varphi$.

For the sake of brevity we will often be writing $\varphi(a, b)$ as $a \cdot b$, or, more succinctly, as $ab$. (In the case of abelian groups we shall instead write $a+b$, although this decision is only a matter of convention.) This shorthand also extends to calling the binary operation ‘multiplication’ (or ‘addition’ in the case of abelian groups) without implying that the operation is commutative or is the usual multiplication (or addition) of numbers. We will also be saying that ‘$G$ is a group’ instead of ‘$G$ is a group under $\varphi$’, unless ambiguity forces us to be more precise.

We can describe the structure of a finite group by arranging all possible products in the group in a grid, with the factors of each product on either end of the grid’s rows and columns. This grid is called the Cayley table after British mathematician Arthur Cayley, who introduced it in his paper On the Theory of Groups as depending on the Symbolic Equation $\theta^n = 1$ (1854). Cayley tables are particularly useful for non-abelian groups, where the product $ab$ is not necessarily equal to $ba$. Some examples of Cayley tables are shown below

• Cayley table for $V_4$.

  $\times$	$e$	$a$	$b$	$c$
  $e$	$e$	$a$	$b$	$c$
  $a$	$a$	$e$	$c$	$b$
  $b$	$b$	$c$	$e$	$a$
  $c$	$c$	$b$	$a$	$e$

• Cayley table for $Q_8$.

  $\times$	$e$	$a$	$a^2$	$a^3$	$b$	$ab$	$a^2b$	$a^3b$
  $e$	$e$	$a$	$a^2$	$a^3$	$b$	$ab$	$a^2b$	$a^3b$
  $a$	$a$	$a^2$	$a^3$	$e$	$ab$	$a^2b$	$a^3b$	$b$
  $a^2$	$a^2$	$a^3$	$e$	$a$	$a^2b$	$a^3b$	$b$	$ab$
  $a^3$	$a^3$	$e$	$a$	$a^2$	$a^3b$	$b$	$ab$	$a^2b$
  $b$	$b$	$a^3b$	$a^2b$	$ab$	$a^2$	$a$	$e$	$a^3$
  $ab$	$ab$	$b$	$a^3b$	$a^2b$	$a^3$	$a^2$	$a$	$e$
  $a^2b$	$a^2b$	$ab$	$b$	$a^3b$	$e$	$a^3$	$a^2$	$a$
  $a^3b$	$a^3b$	$a^2b$	$ab$	$b$	$a$	$e$	$a^3$	$a^2$

2. Homomorphisms and subgroups

In general, our interest in groups is usually not on the groups themselves, but rather on the maps between them. We will be interested in maps that preserve the group structure, which we call homomorphisms.

A more ambiguous, though admittedly less cumbersome, way of saying this is as follows: $\Phi$ is a homomorphism if $$\Phi(ab) = \Phi(a)\Phi(b).$$ In this alternative notation, we do not distinguish between the multiplication in $G$ and the multiplication in $H$, and instead rely on context to determine which group operation we are referring to.

A (group) homomorphism that is injective as a map between sets is called a monomorphism. A homomorphism that is surjective as a map between sets is called an epimorphism. A homomorphism that is both a monomorphism and an epimorphism is called an isomorphism. If there exists an isomorphism between $G$ and $H$, then we say that $G$ and $H$ are isomorphic and write $G \cong H$.

A homomorphism from a group $G$ to itself is called an endomorphism. An endomorphism that is also an isomorphism is called an automorphism.

3. Cyclic groups

4. Cosets and Lagrange's theorem

We introduce the notion of cosets, both as the congruence classes modulo a subgroup and geometrically as the translation of a subgroup by an element of the group. Through this we will be able to prove Lagrange's theorem, a fundamental result in group theory, which will help us describe the structure of finite groups.

The next result is essential in linking the algebraic and geometric interpretations of cosets.

The equivalence classes of $G$ under right (resp. left) congruence modulo $H$ are called the right (resp. left) cosets of $H$ in $G$. We call the element $a \in G$ the coset representative of $Ha$ (and $aH$).

5. Normal subgroups and quotient groups

The next result provides alternative characterizations of normal subgroups.

We call $\pi$ the canonical projection of $G$ onto $G / N$.

6. Symmetric, alternating, and dihedral groups

The identity permutation $\iota$ is the permutation such that $\iota(\xi) = \xi$ for all $\xi \in \Omega$. The product $\sigma\tau$ of two permutations $\sigma, \tau \in S_n$ is the composition $\sigma \circ \tau$. Observe that $\sigma\tau$ is also a permutation of $\Omega$ since the composition of two bijections is a bijection. The product of any permutation with the identity permutation is the permutation itself. Moreover because each $\sigma \in S_n$ is a bijection, there exists a unique permutation $\sigma^{-1}$ such that $\sigma\sigma^{-1} = \sigma^{-1}\sigma = \iota$. We call $\sigma^{-1}$ the inverse of $\sigma$ and is the cycle $$ (\omega_r\ \omega_{r-1}\ \ldots\ \omega_1) = (\omega_1\ \omega_r\ \omega_{r-1}\ \ldots\ \omega_2). $$ Therefore we see that $S_n$ is a group under the product (i.e., composition) of permutations.

Observe that $\sigma\tau = \tau\sigma$ is not necessarily true.

7. Group actions

8. Sylow theorems

9. Abelian groups^*

We first introduce the notion of a free group, and use it to describe groups in terms of generators and relations.

A word on a set $X$ is a sequence of the form $$ a_1, a_2, \ldots, $$ where each $a_i \in X \cup X^{-1} \cup \{1\}$ (here the set $X^{-1}$ is a set disjoint from $X$ such that $|X| = |X^{-1}|$ and whose elements we denote by the bijection $x \mapsto x^{-1}$ and $\{1\}$ is a singleton disjoint from both $X$ and $X^{-1}$, i.e., the choice of $1$ though suggestive is arbitrary) and such that for some $n \in \mathbb{N}$, $a_i = 1$ for all $i > n$. We call $n$ the length of the word and the elements $a_1, a_2, \ldots$ letters. The sequence $$ 1, 1, \ldots $$ is called the empty word and is denoted by $1$.

We say that a word is reduced provided that

1. $a_i = x$ implies $a_{i+1} \neq x^{-1}$ for all $i \in \mathbb{N}$ (i.e., no letter is followed by its inverse)
2. $a_k = 1$ implies $a_i = 1$ for all $i > k$

Every reduced word is thus of the form $$ a_1^{\lambda_1} a_2^{\lambda_2} \cdots a_n^{\lambda_n}, 1, 1, \ldots, $$ where $a_i \in X$ and $\lambda_i = \pm 1$ for all $i = 1, \ldots, n$. For the sake of brevity we shall hereafter denote this word as $$ a_1^{\lambda_1} a_2^{\lambda_2} \cdots a_n^{\lambda_n}. $$ We denote the set of all reduced words on $X$ by $F_X$.

10. Nilpotent and solvable groups^*