Group Extension Example

Question

I am trying to find two distinct group extension of $Z_2$ by $Z_3$. One "natural" extension that I found was $(Z_3,Z_2)$. That is $0 \rightarrow Z_3\rightarrow (Z_3,Z_2)\rightarrow Z_2\rightarrow 0$. However, I am having difficulty in finding another one. Any hints?

Answer 1

There are two (split) extensions, depending on which homomorphism you choose $$ \Bbb{Z}_2 \to \operatorname{Aut} \Bbb{Z}_3 \cong C_2. $$ (I prefer the cyclic group notation $C_2$ for automorphisms that are written multiplicatively, although, of course, $C_2 \cong \Bbb{Z}_2$ as abstract groups.)

If you choose the trivial homomorphism, then you get the direct product. Whereas, if you choose the isomorphism, then you get the semidirect product that is isomorphic to $D_3 \cong S_3$, the symmetries of an equilateral triangle. (The first map embeds $\Bbb{Z}_3$ as the subgroup $A_3$ of rotations and the second projects onto the sign of the permutation.)