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 December 17th, 2014, 11:07 PM #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 09:57 AM.
 January 25th, 2015, 06:37 AM #2 Senior Member     Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 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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