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December 17th, 2014, 11:07 PM   #1
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Question abstract algebra q1

i have a question and thanks everybody for answer
if p and q are Prime number and o(G)=pq and G have normal subgroups with p and q degree prove that ;
G is Cyclic group

thanks very much for your time

Last edited by greg1313; December 18th, 2014 at 09:57 AM.
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January 25th, 2015, 06:37 AM   #2
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Math Focus: Abstract algebra
$G$ has $p-1$ elements of order $p$ and $q-1$ elements of order $q$. Since $pq>(p-1)+(q-1)+1$ it therefore must have an element of order $pq$, say $x$. It follows that $G$ is cylic, since $x^q$ belongs to the Sylow $p$-subgroup and $x^p$ belongs to the Sylow $q$-subgroup.
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