Math Help - Homomorphism Final

1. Homomorphism Final

heres my final one. thnx.

Show that $A_4$, the group of even permutations on 4 letters, is a semidirect product:
$A_4 \cong (C_2 \times C_2) \rtimes_{\varphi} C_3$

and describe explicitly the associated homomorphism:
$\varphi : C_3 \rightarrow Aut(C_2 \times C_2)$

thnx for help on the previous posts.
any help here and ill attempt the rest myself

kind regards
x

2. Originally Posted by joanne_q
Show that $A_4$, the group of even permutations on 4 letters, is a semidirect product:
$A_4 \cong (C_2 \times C_2) \rtimes_{\varphi} C_3$ and describe explicitly the associated homomorphism: $\varphi : C_3 \rightarrow Aut(C_2 \times C_2)$
The group A_4 consists of the identity, eight 3-cycles, and three products of two transpositions, namely (12)(34), (13)(24) and (14)(23). These three products of transpositions, together with the identity, form a normal subgroup isomorphic to $C_2 \times C_2$. The 3-cycles have order 3, of course, so if you take one of them, say (234), it generates a subgroup isomorphic to $C_3$, which acts by conjugation on the normal subgroup. Now all you have to do is to check that the whole group can be identified with the semidirect product of the normal subgroup by that action of $C_3$.