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I am a student at CMU, enrolled in the Abstract Algebra class.

I'm having trouble with a few problems, see if you can figure them out.

Show that for every subgroup $J$ of $S_n|n\geq 2$, where $S$ is the symmetric group, either all or exactly half of the permutations in $J$ are even.

Consider $S_n|n\geq 2$ for a fixed $n$ and let $\sigma$ be a fixed odd permutation. Show that every odd permutation in $S_n$ is a product of $\sigma$ and some permutation in $A_n$.

Show that if $\sigma$ is a cycle of odd length, then $\sigma^2$ is a cycle

Thanks!

Mary

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# Abstract Algebra Problems

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