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Math Help - Prove G is a solvable group

  1. #1
    r45
    r45 is offline
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    Prove G is a solvable group

    Hi there,

    I am trying to solve this problem:

    Suppose G is a finite group, and the order of G is {p_1}^2 {p_2}^2, where p_1,p_2 are primes such that {p_1}^2 < p_2. Show that G is a solvable group.

    I've tried various approaches through Sylow groups and normal subgroups etc but haven't been able to do it, can anyone lend a hand?

    Many thanks!
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  2. #2
    Member Black's Avatar
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    Let's look at the number of Sylow p_2-subgroups. By the Sylow theorems, since n_{p_2}=kp_2+1 and n_{p_2}|p_1^2, n_{p_2} must be 1. Thus, the Sylow p_2-subgroup, P is normal in G. Now consider

    1 \unlhd P \unlhd G.

    Any group of order p^2 (for p prime) is abelian, and since |G/P|=p_1^2, G/P is abelian.
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