Knowlpedia

Proper Subgroup

A subgroup that is strictly smaller than the whole group

Let GG be a . A proper subgroup of GG is a H≤GH\le G such that H≠GH\ne G, equivalently HH is a of GG.

Examples
  • 2Z2\mathbb{Z} is a proper subgroup of Z\mathbb{Z}.
  • AnA_n is a proper subgroup of SnS_n for n≥2n\ge 2.
  • The trivial subgroup {e}\{e\} is proper whenever GG is nontrivial.
Remarks

A nontrivial group is simple precisely when its only normal subgroups are the trivial subgroup and the whole group.