Let $G$ be a non-abelian group of order $p^2q$ and $p> q$.
i) $p \equiv 1 \mod q$ or $p \equiv -1 \mod q$;
ii) The number of elements of order $pq$ in group $G$ is $0$ or $p(p-1)(q-1)$.
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Hint: http://en.wikipedia.org/wiki/Sylow_theorems#Sylow_theorems – Metin Y. Jun 22 '13 at 21:53
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2Welcome on math.SE. For potential answerers it is helpful to include the answers to the following questions in your question: What is the question? Where did you encounter it? What have you tried on it? – Julian Kuelshammer Jun 22 '13 at 22:03
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I need to make proff this teorem.I did not know how to start it because it is non abelian. how do I? – jammie Jun 23 '13 at 04:22
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These groups are discussed in some detail at http://www.math.lsa.umich.edu/~idolga/594Sylow.pdf although some more advanced concepts are used in the exposition. A related question is discussed at http://math.stackexchange.com/questions/57938/application-of-the-sylow-theorems-to-groups-of-order-p2q Also of possible interest is http://math.berkeley.edu/~ogus/Math_113_08/supplements/solsylow.pdf – Gerry Myerson Jun 23 '13 at 12:09
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And there's also http://www.math.purdue.edu/~lipman/5532011/order-p%5E2q.pdf And finally, http://www.math.niu.edu/~rusin/known-math/99/pqq – Gerry Myerson Jun 23 '13 at 12:14