Knowlpedia

Subgroup

A subset of a group that is itself a group under the same operation

Let GG be a with operation ⋅\cdot. A subgroup of GG is a H⊆GH\subseteq G such that:

  1. e∈He\in H (where ee is the identity of GG),
  2. for all a,b∈Ha,b\in H we have a⋅b∈Ha\cdot b\in H,
  3. for all a∈Ha\in H we have a−1∈Ha^{-1}\in H.
Examples
  • For n∈Zn\in\mathbb{Z}, the set nZ={nk:k∈Z}n\mathbb{Z}=\{nk:k\in\mathbb{Z}\} is a subgroup of (Z,+)(\mathbb{Z},+).
  • The AnA_n is a subgroup of SnS_n.
  • The set of diagonal invertible matrices is a subgroup of the group of invertible matrices under multiplication.
Equivalent characterizations

Equivalently, HH is a subgroup iff it is nonempty and closed under the one-step test ab−1∈Hab^{-1}\in H; see the .

Remarks

Subgroups are the basic inputs for and for size constraints in finite groups via .