Question

If H is a subgroup of (g,.) prove that HH=H.

Answer 1

Step-by-step explanation:

We know that H<GH<G. We have

H−1={h−1:h∈H}.

H−1={h−1:h∈H}.

Let x∈H−1x∈H−1. Then x=h−1x=h−1 for some h∈Hh∈H. Because H<GH<G, we have h−1∈Hh−1∈H. Thus, x∈Hx∈H. Hence, H−1⊂HH−1⊂H.

Let y∈Hy∈H. Then y−1∈Hy−1∈H. Hence, y=(y−1)−1∈H−1y=(y−1)−1∈H−1. Thus, H⊂H−1H⊂H−1.

Done.Hope it helps.