Thread: 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?

Thank you in advance

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?

Thank you in advance

let the Klein 4-group. we know that and now let then obviously and since

is a group of order 3, we must have therefore where as usual, is defined by: