## Groups

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

1. For all $x,y\in G$, $x\cdot y \in G$ (closure).

2. There exists an identity element $1\in G$ with $x\cdot 1 = 1\cdot x = 1$ for all $x \in G$ (identity).

3. For all $x,y,z \in G$ we have $(x y)z = x(y z)$ (associativity).

4. 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:

1. Closure.

2. Associativity.

3. Right and left cancellation, namely $a x = b x \implies a = b$ and $y a = y b \implies a = b$.

We shall restrict our attention to finite groups for now.

A homomorphism between two groups $G, H$ is a map $f:G\rightarrow H$ 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 straightforward.∎

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 $H$.