Group

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

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

Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms .

Examples:

• $(\mathbb{Z},+)$ is a group (identity $0$, inverse of $n$ is $-n$).
• $(\mathbb{Q}^{\times},\times)$ is a group (nonzero rationals under multiplication).
• The symmetric group $S_n$ of permutations of $\{1,\dots,n\}$ is a group under composition.
• The set of invertible $n\times n$ real matrices is a group under multiplication (often denoted $GL_n(\mathbb{R})$).