Question

if H is a subgroup of G then HH=H​

Answer 1

Answer:

Step-by-step explanation:

H H=H  

We will show that HH ⊆  H  

Then x  ∈  H H  

Then x  =  ab where a  ∈  H and b  ∈  H  

H H  =  H and H − 1 =H

I already mentioned  that HH=H.

We know that H < G.

We have

H−1={h−1:h∈H}.

Let x ∈ H − 1.

Then x = h − 1 for some h ∈ H. Because H < G, we have h − 1 ∈ H. Thus, x ∈ H.

Hence, H−1⊂H.

Let y ∈ H.

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