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Math Help - subgroup in A4 question

  1. #1
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    subgroup in A4 question

    There is only one subgroup of order 4 in A4 (Alternating group of degree 4)(This subgroup is (1), (12)(34), (13)(24), (14)(23)). Why does this imply that this subgroup must be a normal subgroup in A4? Generalize to arbitrary finite groups.
    I thought about using the fact that for some g that is an element of a group G, |gH^-1g^-1|=|H| (the order of the subgroup H equals each of its conjugates orders) in the whole group, I am not sure were to go from here. Can some help me on this one at all?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by wutang View Post
    There is only one subgroup of order 4 in A4 (Alternating group of degree 4)(This subgroup is (1), (12)(34), (13)(24), (14)(23)). Why does this imply that this subgroup must be a normal subgroup in A4? Generalize to arbitrary finite groups.
    I thought about using the fact that for some g that is an element of a group G, |gH^-1g^-1|=|H| (the order of the subgroup H equals each of its conjugates orders) in the whole group, I am not sure were to go from here. Can some help me on this one at all?
    Theorem: Let G be a group and N\leqslant G be the only group with order |N|=n. Then, N\unlhd G

    Proof: Define \theta_g:N\to gNg^{-1} by n\mapsto gng^{-1}. Clearly this is a bijection. But, notice that geg^{-1}=e\in gNg^{-1}, gng^{-1},gn'g^{-1}\in gNg^{-1}\implies (gng^{-1})(gn'g^{-1})=gnn'g^{-1}\in gNg^{-1} and gng^{-1}gn^{-1}g^{-1}=gnn^{-1}g^{-1} and since n\in N\implies n^{-1} we see that gng^{-1}\in gNg^{-1}\implies gn^{-1}g^{-1}\in gNg^{-1}. Thus, gNg^{-1}\leqslant G. But, since \theta_g was a bijection we know that |gNg^{-1}|=n and since by assumption there was only one subgroup of that order it follows that N=gNg^{-1}. The conclusion follows.
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    could you explain this to me again, I am not getting it still.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by wutang View Post
    could you explain this to me again, I am not getting it still.
    N is a subgroup of order n...the ONLY subgroup of order n. But, given any g\in G, gNg^{-1} is a subgroup of order n. Since there is ONLY ONE subgroup of order n they have to be the same subgroup. Thus, N=gNg^{-1}. That's the definition of normality.
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  5. #5
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    so since are subgroup of order 4 in A4 is the only subgroup of order 4 in A4 it must be normal because, Iam still lost, I thought I had it but I don't! This sucks.
    Last edited by wutang; April 8th 2010 at 09:35 PM.
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  6. #6
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    could you explain specifictly for order 4 in A4 for me? I think I would get it if I just understood that one example.
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  7. #7
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    never mind, I got it finally!!!!
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