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Math Help - Using Sylow to solve the following:

  1. #1
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    Using Sylow to solve the following:

    Dear All,

    I am stuck on a partucular question which asks the following:

    Let G be a group of order pq where p and q are distinct primes. Prove that G is soluble:

    I beileve that one method is to use Sylow but I am having trouble solving it. Can someone please assist.

    Thanks.
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  2. #2
    Senior Member roninpro's Avatar
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    Hello.

    Recall that a group G is solvable if we can find a series of groups 0=G_0\leq G_1\leq \ldots \leq G_n=G such that G_i is normal in G_{i+1} and G_{i+1}/G_i is abelian.

    Suppose that a group G is a group of order pq, where p and q are distinct primes (and without loss of generality, suppose p<q. By the first Sylow theorem, there are subgroups of G of order q. By the third Sylow theorem, the number of of these subgroups n_q divides p and satisfies n_q\equiv 1\pmod{q}. We can conclude that n_q=1. Take P to be that subgroup. By the second Sylow theorem, P is normal in G. We can write the series 0\leq P\leq G, noting that P/0 is abelian since it is prime order p and G/P is abelian since it is prime order q.

    This proves it.
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  3. #3
    Member Black's Avatar
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    Nov 2009
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    Alternatively, consider

    1 \unlhd Z(G) \unlhd G.

    By Lagrange's theorem, |Z(G)| \in \{1,p,q,pq\}. If Z(G)=1, \, pq, then we're done, so let |Z(G)|=p.
    Then |G/Z(G)|=q. Since q is prime, G/Z(G) is cyclic. The case for |Z(G)|=q is similar.
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