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Thread: Simple groups that are maximal subgroups of G.

  1. #1
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    Simple groups that are maximal subgroups of G.

    Hi:
    I'll make a straightforward question: Let $\displaystyle G$ be group and $\displaystyle S \leq T \leq G$, these groups satisfying the following conditions:

    (a) $\displaystyle T \unlhd G$.
    (b) $\displaystyle T$ is simple.
    (c) $\displaystyle T$ is a maximal subgroup of$\displaystyle G$.

    Is there any other piece of information that can be deduced from these three facts? In particular, can I infer $\displaystyle S \unlhd G$?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Simple groups that are maximal subgroups of G.

    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    I'll make a straightforward question: Let $\displaystyle G$ be group and $\displaystyle S \leq T \leq G$, these groups satisfying the following conditions:

    (a) $\displaystyle T \unlhd G$.
    (b) $\displaystyle T$ is simple.
    (c) $\displaystyle T$ is a maximal subgroup of$\displaystyle G$.

    Is there any other piece of information that can be deduced from these three facts? In particular, can I infer $\displaystyle S \unlhd G$?
    Well, no, because $\displaystyle T$ is simple. If $\displaystyle S\leq T\leq G$ and $\displaystyle S\lhd G$ then $\displaystyle S\lhd T$...
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  3. #3
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    Re: Simple groups that are maximal subgroups of G.

    That's a pity, because I'm in the process of solving a problem, half of which, it seems, I have already solved, and I thought I could extract aditional info from those facts. About the problem it would be unfair to speak because I've already posted in MHF stating its hypothesis (and the propositions I have arrived at up to now). Thanks for your reply.
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