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Math Help - permutation and semidirect product

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    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 V=\{(1), (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3) \}, the Klein 4-group. we know that V \lhd A_4 and V \simeq C_2 \times C_2. now let K=<(1 \ 2 \ 3) >. then obviously K \simeq C_3, \ V \cap K = \{(1)\}, and since A_4/V

    is a group of order 3, we must have A_4/V \simeq K. therefore A_4=V \rtimes_{\varphi} K, where \varphi: K \longrightarrow \text{Aut}(V), as usual, is defined by: \varphi(k)(v)=kvk^{-1}, \ \forall k \in K, \ \forall v \in V.
    Last edited by NonCommAlg; February 9th 2009 at 01:31 PM.
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