based on group theory. Show that for elements g & h of a group, the following are equivalent:?

Question

(i) gh = hg

(ii) (gh)^2 = g^2h^2

(iii) (gh)^(-1) = g^(-1)h^(-1)
I'm not really sure how to prove that one side equals the other side or where to start

Answer 1

To show that two statements a and b are equivalent, you assume a and prove b; then you assume b and prove a.

To show that three statements i ii and iii are equivalent, one way is to assume i and prove ii, then assume ii and prove iii, and then assume iii and prove i. Of course, you don't have to do them in that order.

i->ii:
Suppose gh = hg.
Then, (gh)^2 = ghgh = gghh (because hg = gh)
gghh = g^2h^2

The rest, ii-> iii and iii-> i, are similarly easy.