Hi Everyone,

My problem is stated as: "Determine the groups G such that $\displaystyle \{(g,g^{-1})\mid g \in G\}$ is a subgroup on $\displaystyle G\times G$." To me, appears that if G is a group, then this should always define a subgroup.

Going through the subgroup test:

Identity: All of the resulting subgroups will have an identity, namely $\displaystyle (1_G,1_G)$.

Units: All elements are invertible. For any pair $\displaystyle (g,g^{-1})$, the inverse is $\displaystyle (g^{-1},g)$, which also exists in the subgroup.

Closed: For any $\displaystyle a,b\in G$, we would have $\displaystyle (a,a^{-1})(b,b^{-1})=(ab,a^{-1}b^{-1})$, which should also exist in the subgroup.

Is there something I'm missing? The question seems to imply that in some cases, this subset will not be a subgroup, but I don't see when that would occur.

Thanks,

Peter