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Math Help - Help with a proof - the commutator subgroup

  1. #1
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    Help with a proof - the commutator subgroup

    I need to prove the following:

    Let G be a group.
    If H<G s.t. S<H, then H\triangleleft G (S is the commutator subgroup).
    I just need a lead here...

    Thank you in advance.
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  2. #2
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    Re: Help with a proof - the commutator subgroup

    Quote Originally Posted by Stormey View Post
    I need to prove the following:
    Let G be a group. If H<G s.t. S<H, then H\triangleleft G (S is the commutator subgroup).
    You really should define all the terms and symbols in your question.
    Why do you think the notation is standard?
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  3. #3
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    Re: Help with a proof - the commutator subgroup

    Stormey,
    I think the commutator subgroup of G is universally denoted by G'. The easiest way to answer your question is to work in the factor group G/G'. However a direct approach is:
    Let h be in H and g in G. Then [g,h-1]=g-1hgh-1 is in H and so g-1hg is in H.
    Thanks from Stormey
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    Re: Help with a proof - the commutator subgroup

    Quote Originally Posted by Plato View Post
    You really should define all the terms and symbols in your question.
    Why do you think the notation is standard?
    Sorry.
    here it is again with standard notations:

    I need to prove the following:

    Let G be a group.
    If H\leq G s.t. H'\leq H, then H\triangleleft G.
    I just need a lead here...

    Thank you in advance.

    Quote Originally Posted by johng View Post
    Stormey,
    I think the commutator subgroup of G is universally denoted by G'. The easiest way to answer your question is to work in the factor group G/G'. However a direct approach is:
    Let h be in H and g in G. Then [g,h-1]=g-1hgh-1 is in H and so g-1hg is in H.
    Thanks for the help.
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