The question I am working with is
Let $(G,\star)$ be a group. Define the map $\star ':G \times G \rightarrow G$ by $a\star 'b =b\star a$ for $a,b \in G$.
I was able to prove that $(G, \star ') $ is in fact a group, but after looking around the web,and reading through my text, I am having overall troubles finding how to prove $(G, \star ') $ is isomorphic to $(G,\star)$.
I know the definitions and that I need to show $\varphi(ab) = \varphi(a)\varphi(b)$ and that $\varphi$ is bijective, but am stuck on how to actually do it.
The group you define is known as the opposite group of $G$, and is often denoted by $G^{\operatorname{opp}}$. Every group $G$ is canonically isomorphic to its opposite group by the map $$G\ \longrightarrow\ G^{\operatorname{opp}}:\ g\ \longmapsto\ g^{-1}.$$