I need some help figuring out some qualities of even permutation groups. Consider $E_n$ to be a subset of the bijection set $S_n$ (bijections over $[n]$) that consists of all even permutations. I want to show that there are no trivial normal subgroups of $E_5$ (as in, if $E_5$ has a normal subgroup, it must be either $(e)$ or $E_5$ itself). I was wondering if it might be best to use a version of Sylow's Theorem and then use the properties of the order of $E_5$ to suggests something about the quality of $E_5$'s subgroups. That being said, I cannot be sure if this is an appropriate method.
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4The usual notation is $A_n$, the alternating group. – lhf Jan 08 '16 at 01:27
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The simplest approach I know uses the Class Equation. Are you familiar with that result? – John Brevik Jan 08 '16 at 01:28
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Where you wrote "no trivial normal subgroups" did you mean "no non-trivial normal subgroups"? ${}\qquad{}$ – Michael Hardy Jan 08 '16 at 03:49