Let $G$ be a group of order $pqr$, where $p$, $q$ and $r$ be three distinct primes. By Cauchy's theorem there exist three elements, $a$, $b$ and $c$, whose orders are $p$, $q$ and $r$, respectively. If the subgroup generated by $a$ and $b$ is the whole group, then I wonder if it is possible that there exists a proper normal subgroup of $G$.
Can $G$ of order $pqr$ be simple if it's generated by elements of orders $p,q$?
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group-theory
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0I edited your question to make it a bit clearer; and also presumably you are wondering if there is a *proper* normal subgroup, not just any normal subgroup ($G$ and {$e$} are normal in $G$, after all). – 2010-10-05
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0More precisely, i wonder whether the group is simple. – 2010-10-05
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0@0592: that's exactly the same as asking if there is a proper normal subgroup. The point is, you asked if there is a normal subgroup: there *always* is a normal subgroup (the trivial one). – 2010-10-05
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0you are right, "proper" cannot be missed here. Thanks for your remind. – 2010-10-05