Groups

Definition

A group is a set $G$ together with a binary operation $\cdot :G\times G\to G$ satisfying:

1. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for all $a,b,c\in G$
2. Identity: There exists $e\in G$ such that $e\cdot a=a\cdot e=a$ for all $a\in G$
3. Inverses: For each $a\in G$, there exists $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$

Examples

• $(\mathbb{Z},+)$ — the integers under addition
• $(S_n,\circ )$ — the symmetric group on $n$ elements
• $({\text{GL}}_n(\mathbb{R}),\cdot )$ — invertible $n\times n$ matrices

Subgroups

A subset $H\subseteq G$ is a subgroup if it is closed under the group operation and inverses. We write $H\le G$.