1. permutation and semidirect product

1) Show that A_4 ( the group of even permutations on 4 letters) is a semi direct product: A_4 =~ (C_2 x C_2) ⋊φ C_3

2) Describe explicitly the associated φ : C_3 --> Aut(C_2 x C_2)

Here is my sketch:

First I am going to find the permutations of A_4. I think there are 12 elements of permutation in A_4 then find all the homomorphisms of (C_2 x C_2) ⋊φ C_3

But I dont know if this is the right way to do this question. Can any body help please?

2. Originally Posted by knguyen2005
1) Show that A_4 ( the group of even permutations on 4 letters) is a semi direct product: A_4 =~ (C_2 x C_2) ⋊φ C_3

2) Describe explicitly the associated φ : C_3 --> Aut(C_2 x C_2)

Here is my sketch:

First I am going to find the permutations of A_4. I think there are 12 elements of permutation in A_4 then find all the homomorphisms of (C_2 x C_2) ⋊φ C_3

But I dont know if this is the right way to do this question. Can any body help please?

let $\displaystyle V=\{(1), (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3) \},$ the Klein 4-group. we know that $\displaystyle V \lhd A_4$ and $\displaystyle V \simeq C_2 \times C_2.$ now let $\displaystyle K=<(1 \ 2 \ 3) >.$ then obviously $\displaystyle K \simeq C_3, \ V \cap K = \{(1)\},$ and since $\displaystyle A_4/V$
is a group of order 3, we must have $\displaystyle A_4/V \simeq K.$ therefore $\displaystyle A_4=V \rtimes_{\varphi} K,$ where $\displaystyle \varphi: K \longrightarrow \text{Aut}(V),$ as usual, is defined by: $\displaystyle \varphi(k)(v)=kvk^{-1}, \ \forall k \in K, \ \forall v \in V.$