Let be a finite group such that . If for the elements it holds that

then prove that is abelian.

**Solution**

Let such that . If then the order of would be . This would immediately imply that the order of would divide the order of the group . This is an obscurity due to the data of the exercise. Thus . As we conclude that the mapping is an group homomorphism.

Therefore forall we have or equivelantly . Taking advantage of the last relation we get that:

as well as

The last two relations hold for all . Thus, for all it holds that:

which in turn implies

and since is we eventually get proving the claim that is abelian.