Hi, need quick help with these questions, have exams on Tuesday

1) Let G be a group. Prove that

I have been able to prove it, first with n=1, then assuming it holds for n=k and proving for n=k+1 and n=-m for some +ve int m

But the lecturer used n=0 for the first one,

For n=0, he got and for RHS

I dont understand how or

2) Let G be a group and a is an element of G. Prove

For second question i have no idea at all, what is o(a) ??

Third question i have been able to do, needs to verify if its corect

3) A relation R is defined onZby aRb iff a-b is divisible either by 5 or by 7(a,b inZ). Is R an equivalence relation onZ? Justify your answer.

For reflexive, 5|a-a or 7|a-a = 5|0 or 7|0

Hence aRa.

For symmetric, aRb implies 5|a-b or 7|a-b = 5|b-a or 7|b-a which implies bRa.

For transitivity, aRb and bRc implies

5|a-b or 7|a-b and 5|b-c or 7|b-c

= 5|a-b+b-c or 7|a-b+b-c

= 5|a-c or 7|a-c

=aRc

Hence R is an equivalence relation onZ