Proving Group Properties of $\overline{G}$

[Punkyc7]

So I have this group G. Then I make set $\overline{G}$ with a new operation <$\overline{G}$,*> a*b=ba. Show $\overline{G}$ is a group,So showing associativity is easy. But I am unsure how to show the identities and the inverse.

I want to same the identity is 1 but since we don't have any numbers I have a feeling that could be wrong and for the inverse just a^(-1). How do you know what is going to be what if you don't have a specific structure like the reals or the integers to work with?

Just curious if $\overline{G}$$\subseteq$ G then would $\overline{G}$ retain G identity?

 

[lippyka]

No, \overline{G} does not necessarily retain the identity of G. The operation in \overline{G} is different from the operation in G, so the identity element may also be different. To show that \overline{G} is a group, you need to show that it has an identity element and inverse elements for each element.