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Jack Schmidt  
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 More options Jul 24, 6:27 am
Newsgroups: sci.math
From: Jack Schmidt <Jack.Schmidt.SciM...@gmail.com>
Date: Wed, 23 Jul 2008 18:27:29 EDT
Local: Thurs, Jul 24 2008 6:27 am
Subject: Re: - - products of subgroups
crossedproduct asked at http://mathforum.org/kb/thread.jspa?messageID=6305486

> Suppose P and Q subgroups of a group G such that P
> and QP are normal in G.

> If g is any element of G, denote by x^g the conjugate
> gxg^{-1}.

> If p is in P and q is in Q, then for any g in G,

> (qp)^g = (q^g)(p^g) is in QP,

> since QP is normal. But p^g is in P, since P is
> normal.

> My question is: why does it follow that q^g must lie
> in either P or Q ?

There are counterexamples with G of order 8.

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