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Math Help - Finitely generated

  1. #1
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    Finitely generated

    A group H is called finitely generated if there is a finite set A s.t. H=<A>

    Prove that every finite group is finitely generated.

    <A>=\bigcap_{\begin{cases}A\subset H\\H\leq G\end{cases}}H

    I don't know how to set that up with out the { and make it smaller under the intersection.

    Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Finitely generated

    Quote Originally Posted by dwsmith View Post
    A group H is called finitely generated if there is a finite set A s.t. H=<A>

    Prove that every finite group is finitely generated.

    <A>=\bigcap_{\begin{cases}A\subset H\\H\leq G\end{cases}}H

    I don't know how to set that up with out the { and make it smaller under the intersection.

    Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?
    I'm not really sure what you're trying to get at. The idea is that a group is generated by a subset if (roughly) all elements in the group are just products of things in the set, and things whose inverse is in the set. What if you had a finite group G, what finite set has the property that every element of G is a product of things in this set?
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  3. #3
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    Opalg's Avatar
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    Re: Finitely generated

    Quote Originally Posted by dwsmith View Post
    A group H is called finitely generated if there is a finite set A s.t. H=<A>

    Prove that every finite group is finitely generated.

    <A>=\bigcap_{\left\{{A\subset H}\atop{H\leq G}\right.}H

    I don't know how to set that up with out the { and make it smaller under the intersection.

    Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?
    You can just take A=G.

    Edit. Sorry Drexel28 (shouldn't that now be Maryland28?), I didn't see your reply.

    Edit 2. Notice that I can make the { under the intersection smaller.
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