Question

If (G,*) is group, then show that (a-1)-1=a

Answer 1

Step-by-step explanation:

Note that by definition (a−1)−1 and a are both inverse elements of a−1. Since in a group each element has a unique inverse, we can conclude that (a−1)−1=a.

I will now prove that in a group inverse elements are unique.

Suppose b,c∈G such that ba=e=ab and ca=e=ac That is to say, b and c are both inverse elements of a where e is the identity of the group G.

Then we have b=be=b(ac)=(ba)c=ec=c

Therefore b=c and inverse elements are unique.