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 on Z by aRb iff a-b is divisible either by 5 or by 7(a,b in Z). Is R an equivalence relation on Z? Justify your answer.
For reflexive, 5|a-a or 7|a-a = 5|0 or 7|0
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
Hence R is an equivalence relation on Z