I am trying to understand the following problem
When I looked at the question, I felt if $a\in G$, then for every positive integer m, $a^m \in G$. Isn't this true? because a Group should satisfy the closure property?
And then it is given in the question as $a^m b^m = b^m a^m$. Which means that the group is abelian. Where am I going wrong with this argument?
If I am wrong in taking $a^m \in G$, then in the solution given in the book, at many places he used things like assuming $c = (b^n a^m)^x$ and then using $a^m c^m = c^m a^m$.
But according to the question this can be used only when a and c are elements of G. How did the author here decide that c belongs to G?