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Math Help - Subgroup with prime index

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    Subgroup with prime index

    Prove that A_6 has no subgroup of prime index.
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  2. #2
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    Quote Originally Posted by didact273 View Post

    Prove that A_6 has no subgroup of prime index.
    you can prove a much stronger result: if n \geq 5 and H \lneq A_n, then [A_n:H] \geq n. this is a quick result of this well-known fact that for any group G and any subgroup H of G of finite index, there

    exists a normal subgroup N of G contained in H such that [G:N] \mid [G:H]!.
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    Quote Originally Posted by NonCommAlg View Post
    you can prove a much stronger result: if n \geq 5 and H \lneq A_n, then [A_n:H] \geq n. this is a quick result of this well-known fact that for any group G and any subgroup H of G of finite index, there

    exists a normal subgroup N of G contained in H such that [G:N] \mid [G:H]!.
    Can't I deduce from the fact that since A_6 is a simple group, the only possible subgroup is {(1)} and itself, there is no subgroup of index p?
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    Quote Originally Posted by didact273 View Post

    Can't I deduce from the fact that since A_6 is a simple group, the only possible subgroup is {(1)} and itself, there is no subgroup of index p?
    the only possible (proper) normal subgroup is {(1)}. see the definition of a simple group again!
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