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Math Help - One biggest subgroup

  1. #1
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    One biggest subgroup

    Let G be a group and H a proper subgroup which contains every proper subgroup of G. What can be said about G?

    I can't seem to figure out what the question is referring to.
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  2. #2
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    Re: One biggest subgroup

    my guess would be that G is cyclic. let's see if we can prove this.

    suppose H is such a group. since H is proper, there must be some x in G, with x not in H.

    now <x> is a subgroup of G. if it were proper, then by our assumption on H, <x> would be contained in H (since H contains every proper subgroup).

    in particular, x would have to be in H, since x is in <x>. but this contradicts our choice of x.

    hence <x> must not be proper, in other words, <x> = G, so G is cyclic.
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  3. #3
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    Re: One biggest subgroup

    Yes, G is cyclic, but is it possible to have a proper nontrivial H with such property?
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  4. #4
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    Re: One biggest subgroup

    I think it is possible. If G has prime-power order p^a, then the subgroup of order p^{a-1} works (assuming a\ge{2}).

    In fact, I think if G is finite, it has to have prime-power order. If p and q are distinct prime divisors of |G|, then the subgroups of orders \frac{|G|}{p} and \frac{|G|}{q} are contained in H - meaning H cannot be a proper subgroup.

    Also it can't be infinite, since then it would be isomorphic to the integers and the integers don't have a single largest subgroup.
    Last edited by hollywood; July 24th 2012 at 04:46 PM. Reason: must be prime power order
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