Basic Problem on Direct Products

Question

I am trying to prove that if $A$ and $B$ are groups, $A \times B$ is isomorphic to $B \times A.$ I have a feeling that I will want to use the main theorem of direct products, i.e. that if $G N_1N_2...N_n$ is a group and $F = N_1 \times N_2 \times ... \times N_n,$ then $G$ and $F.$ So, I know that $A \times B$ is isomorphic to $AB$, and $B \times A$ is isomorphic to $BA,$ but I am not sure if it would be appropriate to develop a bijective homomorphism between $AB$ and $BA.$ Any suggestions on this problem?

Answer 1

I don't know if there are shortcuts using theorems, but the natural bijection between $A\times B$ and $B \times A$ is $$\phi((a,b))=(b,a)$$ By 'natural' I mean 'following from the observation that the sets $A \times B$ and $B \times A$ are 'the same with just the order of elements swapped'.

You just have to show $\phi$ is a bijection (find an inverse function) and show $\phi((a_1,b_1))\phi((a_2,b_2))=\phi((a_1,b_1)(a_2,b_2))$ to demonstrate isomorphism.

Answer 2

Hint: look at the mapping from $A \times B$ to $B \times A$ give by $(a,b) \mapsto (b,a)$.