Hehe, I'm working through the complete groups books right now, so don't think I ask you all my homework questions... I'm doing alot myself too =).(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

1) If H is a subgroup of [tex]S_n[/tex], and is not contained in [tex]A_n[/tex], show that precisely half of the elements in H are even permutations

2) show that for n>3, all elements of [tex]S_n[/tex], can be written as a product of two permutations, each of which has order 2.

3. The attempt at a solution

1) If there is an element x that's not in [tex]A_n[/tex], then x=yz, where either y or z is odd.

call this element x'. x'=y'z', and either y' or z' is odd again.

If you continue this process, you eventually have to get back to x, because [tex]S_n[/tex] is finite. If you can show this is when you repeated the process n/2 times, you're done... because the other half are the even ones you encountered when you wrote it as a product every time.

But this is a step I'm having problems with. And I realize this argument has loads of holes in it.

2) I have only a vague clue on this one. I thought I might write the element as a product of disjoint cyclic permutations. Now cyclic permutations are of order 2.

because this is grammatically close to what I want I thought it might help =P, but I really don't know how get from here to the point where you have two elements of order 2.

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# Homework Help: Permutation groups

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