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December 18th, 2014, 12:07 AM  #1 
Newbie Joined: Dec 2014 From: all of world is my home Posts: 2 Thanks: 0  abstract algebra q1
hello 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 10:57 AM. 
January 25th, 2015, 07:37 AM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 155 Math Focus: Abstract algebra 
$G$ has $p1$ elements of order $p$ and $q1$ elements of order $q$. Since $pq>(p1)+(q1)+1$ 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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