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 March 21st, 2009, 10:51 AM #1 Newbie   Joined: Mar 2009 Posts: 21 Thanks: 0 Symmetric Groups 1. Show that $S_n=\left< (1 2),(1 2 3 . . . n) \right=>=$ for all $n \geq 2$. 2. Show that if $p$ is prime, $S_p=<\sigma , \tau=>=$ where $\sigma$ is any transposition and $\tau$ is any p-cycle. 3. Show that $\left< (1 3), (1 2 3 4) \right>$ is a proper subgroup of $S_4$. What is the isomorphism type of this subgroup? It seems like if I could do 1, then 2 and 3 would follow pretty easily...I'm not sure.
 March 24th, 2009, 10:22 AM #2 Senior Member   Joined: Dec 2008 Posts: 160 Thanks: 0 Re: Symmetric Groups The only thing you need to show for 1 and 2 is that in ech case there is sequence of operations that exchange any two elements, while keeping the rest.

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