I did a test recently and I struggled on these questions

(1) Prove in any group (x^-1)^-1=x

I said (x^-1)^-1=a so a^-1(x^-1)^-1=e

so (x^-1*a)^-1=e implying x^-1*a=e, implying a=x.

But that's assuming what is to be proved. namely that the inverse of x^-1 is x.

(2) Prove if a and b are conjugate, they have the same order.

My answer. Let order of a=k and let e be the identity.

a=gbg^-1 for some g

a^k=(gbg^-1)^k=(g^-1)^k*b^k*g^k=e

This implies b^k=e so order of b=k. But I don't think that's fully correct because it doesn't show k is the smallest integer such that b^k=e.

(3) decide which of the following are isomorhic, quoting results you use

Z2 xZ6, Z3XZ4, Z2 X Z2 X Z3, Z12. My failure is just ignorance with respect to these standard results. I only knew that Z12 is cyclic. For Zm X Zn to be cyclic, m and n need to be co-prime. 2 and 6 are not co-prime so Z2 X Z6 is not cyclic and so not isomorphic to Z12.

Help much appreciated.