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Math Help - Group Theory-Show no infinite soluble group has a composition series

  1. #1
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    Group Theory-Show no infinite soluble group has a composition series

    Hi.....i'm stuck on the following question;

    Show that no infinite soluble group has a composition series.


    The hint for the question was to try by contradiction.

    So i tried the following;

    Suppose G is an infinite soluble group with a composition series.
    G is soluble so all composition factors are cyclic and of prime order.

    I think there will be a contradiction because G is infinite...but i just don't know

    Thanks for your help
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  2. #2
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    Quote Originally Posted by Louise View Post
    Hi.....i'm stuck on the following question;

    Show that no infinite soluble group has a composition series.


    The hint for the question was to try by contradiction.

    So i tried the following;

    Suppose G is an infinite soluble group with a composition series.
    G is soluble so all composition factors are cyclic and of prime order.

    I think there will be a contradiction because G is infinite...but i just don't know

    Thanks for your help
    You almost have it: there's a (finite) series for G all of which factors are finite cyclic of prime order. But then, since a finite extension of a finite group is itself finite, we see G must be finite...(if G >= G_1 >= G_2 >=....>=G_n= 1 is the cyclic series, then G_n is finite, so G_{n-1} is finite since G_n\,\, and\,\, G_{n-1}/G_n are finite, etc...)

    Tonio
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