Prove the set of even permutation in forms a subgroup of .

How should I start this?

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- Sep 11th 2007, 07:28 PMtttcomraderPermutation subgroup
Prove the set of even permutation in forms a subgroup of .

How should I start this? - Sep 11th 2007, 07:40 PMThePerfectHacker
This is referred to an "alternating group" .

How about you start off and I help you if necessary.

Let be a group and a non-trivial subset of . What makes a subgroup? What conditions must be met?

Now do the same thing here. Show all of these conditions are satisfied. - Sep 11th 2007, 08:20 PMtttcomrader
Okay, let me think more about this...

Suppose that with with a being permutations that can be expressed as even number of 2-cycles, in other words, H contains all of the even permutations in Sn.

Now pick .

Then and

Now, is in H because it is only another expression of even permutations. (Do I need to justify this? Or do we already know that? I know that by the problems that I have done before)

Then is its own inverse, thus satisfied the conditions for H to be a subgroup of Sn.

Am I doing something right here?

Thanks. - Sep 11th 2007, 08:30 PMThePerfectHacker
[QUOTE=tttcomrader;69480]

Quote:

Am I doing something right here?

.

is even and is even then is even. You mentioned that but you did not show it. Why is that? Remember you need to show this to show that is closed.