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

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

    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

    $\displaystyle 1 \unlhd Z(G) \unlhd G.$

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