Knowlpedia

Subgroup Test (one-step)

A nonempty subset H of a group is a subgroup exactly when xy⁻¹ belongs to H for all x,y in H.

One-step subgroup test. Let GG be a and let HH be a nonempty of GG. Then HH is a of GG if and only if

x,y∈H⟹xy−1∈H.x,y\in H\quad\Longrightarrow\quad xy^{-1}\in H.
Remarks

Taking x=yx=y gives the identity, taking x=ex=e gives inverses, and the displayed condition then gives closure under products.