Subgroup

Let $G$ be a group with operation $\cdot$. A subgroup of $G$ is a subset $H\subseteq G$ such that:

1. $e\in H$ (where $e$ is the identity of $G$),
2. for all $a,b\in H$ we have $a\cdot b\in H$,
3. for all $a\in H$ we have $a^{-1}\in H$.

Equivalently, $H$ is a subgroup iff it is nonempty and closed under the one-step test $ab^{-1}\in H$; see the subgroup test . Subgroups are the basic inputs for cosets and for size constraints in finite groups via Lagrange's theorem .

Examples:

• For $n\in\mathbb{Z}$, the set $n\mathbb{Z}=\{nk:k\in\mathbb{Z}\}$ is a subgroup of $(\mathbb{Z},+)$.
• The alternating group $A_n$ is a subgroup of $S_n$.
• The set of diagonal invertible matrices is a subgroup of the group of invertible matrices under multiplication.