Group Theory - Groups

Groups

A group is a set $G$ and a binary operation $\cdot$ such that

For all $x, y \in G$ , $x \cdot y \in G$ (closure).
There exists an identity element $1 \in G$ with $x \cdot 1 = 1 \cdot x = 1$ for all $x \in G$ (identity).
For all $x, y, z \in G$ we have $(x y) z = x (y z)$ (associativity).
For all $x \in G$ there exists an element $x^{- 1}$ with $x x^{- 1} = x^{- 1} x = 1$ (inverse).

If we only have closure and associativity, then we call $G$ a semigroup. If we have closure, associativity and an identity element, we call $G$ a monoid.

If $x y = y x$ for some $x, y \in G$ then we say $x, y$ commute (or are commutative, or permutable). If $x y = y x$ for all $x, y \in G$ then we say $G$ is abelian (or commutative).

Theorem: The following are alternative axioms for defining finite groups:

Closure.
Associativity.
Right and left cancellation, namely $a x = b x \Rightarrow a = b$ and $y a = y b \Rightarrow a = b$ .

We shall restrict our attention to finite groups for now.

A homomorphism between two groups $G, H$ is a map $f : G \to H$ with $f (x) f (y) = f (x y)$ for all $x, y \in G$ . If $f$ is bijective then we call $f$ an isomorphism.

The order of an element $g$ in a group $G$ is the smallest positive integer $k$ such that $g^{k} = 1$ . This must always exist in a finite group.

Theorem: If $x \in G$ has order $h$ , then $x^{m} = 1$ if and only if $h ∣ m$ .

Theorem: If $x \in G$ has order $m n$ , where $m, n$ are coprime, then $x$ can be uniquely expressed in the form $x = u v$ where $u$ has order $m$ and $v$ has order $n$ .

Proof: Find $a, b$ with $a m + b n = 1$ , and pick $u = x^{b n}, v = x^{a m}$ . Uniqueness is not difficult to prove. $▪$

A subset $H$ of $G$ that also satisfy the group axioms is called a subgroup of $G$ . Every group $G$ contains two trivial or improper subgroups, $G$ itself and the group consisting of the identity element alone. All other subgroups are called proper subgroups.

Theorem: A nonempty subset $H$ of $G$ is a subgroup if and only if it is closed under multiplication.

A nonempty subset $H \subset G$ is a subgroup if and only if $H^{2} \subset H$

Lemma: For a subgroup $H$ , for all $h \in H$ we have $h H = H = H h$ .

Corollary: For any set $S \subset H$ we have $S H = H = H S$ .

We can now strengthen a previous statement. A nonempty subset $H \subset G$ is a subgroup if and only if $H^{2} = H$

Theorem: Let $g \in G$ . Then for a subgroup $H$ , we have $g^{- 1} H g$ is also a subgroup of $G$ isomorphic to $H$ .