Knowlpedia

Subgroups are closed under inverses and products

A subgroup contains the identity and is closed under multiplication and inversion

Proposition (Closure properties of subgroups). Let HH be a of a GG. Then:

  1. The identity element ee of GG lies in HH.
  2. If h∈Hh\in H, then h−1∈Hh^{-1}\in H.
  3. If h1,h2∈Hh_1,h_2\in H, then h1h2∈Hh_1h_2\in H.
  4. Consequently, if h1,h2∈Hh_1,h_2\in H, then h1h2−1∈Hh_1h_2^{-1}\in H.
Remarks

Context. The reverse implication (a nonempty subset closed under h1h2−1h_1h_2^{-1} is a subgroup) is packaged as a , so this proposition is the "easy direction" used constantly in computations.