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Math Help - Finite simple groups.

  1. #1
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    Finite simple groups.

    Hi: Problem: Let U be a subgroup of G and 1 < |G:U| <5. Then |G| < 4 or G is not simple.

    If |G:U| = 2 I can use a problem (problem A) that says: Let p be the smallest prime divisor of |G|. Then every subgroup of index p is normal in G. That is because in this case, 2 is a divisor of |G| and it is the smallest prime. For case |G:U| = 3 if |G| is not even then, by problem A I am done. If |G| is even, however, problem A is of no use. The same for |G:U| = 4. Any hint?
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  2. #2
    Member Sylvia104's Avatar
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    Re: Finite simple groups

    There is a result in group theory that says that if n=|G:U| then there is a normal subgroup N of G contained in U such that n divides |G:N| and |G:N| divides n!. (I don't know what this result is called but I've found it useful. ) Thus for n=3, |G:N|=3 or 6 so N\ne1; if |G|>3, then N\ne G either (since N is contained in U and |G:U|=3), showing that G is not simple.

    If |G:U|=4, then 4\mid|G:N| and |G:N|\mid4!=24 \implies G is not simple if |G|>4.

    If |G:U|=2 then U itself is a normal subgroup of G and so G is not simple if |G|>2.
    Last edited by Sylvia104; April 20th 2012 at 05:25 PM.
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  3. #3
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    Re: Finite simple groups.

    Thank you very much Sylvia.
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