The left regular action is defined as follows: Let G be a group and A = G (as sets.) Define $\displaystyle \phi : G \times A \to A: g \cdot a \mapsto ga$ where g is any element of G and a is any element of A. ga is calculated using the group operation in G. The question is to find $\displaystyle ker( \phi )$. The kernal in my text is defined as $\displaystyle ker ( \phi ) = \{g \in G | g \cdot a = a~\forall a \in A \}$.

The only element I can come up with for the kernal is $\displaystyle 1_G$. That seems to little to me. Are there any others? A hint would be appreciated

Thanks.

-Dan