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Let $p$ and $q$ be integer primes such that $p$ divides $q-1$.

(a) Show that there exists a group $G$ of order $p^{2}q$ with generators $x$ and $y$ such that $x^{p^{2}} =1$, $y^{q}=1$, and $xyx^{-1}=y^{a}$, with $1$ the identity element of $G$ and $a$ some integer such that $a\not\equiv 1 \pmod q$ but $a^{p}\equiv 1\pmod q$.

(b) Prove that Sylow $q$-subgroup $S_{q}$ is normal in $G$, $G/S_{q}$ is cyclic and deduce that $G$ has a unique subgroup $H$ of order $pq$.

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    Show your work? We are not here to give answers to your homework.2012-12-29
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    ok, I have just come back,I should say that this is not my homework. Actually this is to show the existence of the semidirect product of two cyclic group of order $p^{2} ,q$, since the Aut of the group of order q is of order q-1, and p divide q-1, so we can find the the homomorphism required for the semidirect product.2012-12-29
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    I am really doing lots of problems recently preparing for the exams, I did try for a long time, but I am not familiar with some definition and theorem, so I have not idea of what is going on, so any hint and even recommendation is help for me. Thanks a lot.2012-12-29

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