Math Is Fun Forum

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

odd permutations form a group...identity

Do the odd permutations of S_n form a group? Explain.

I KNOW that they dont for the reason that the group of odd permutations does not include the identity. I'm not sure how to prove this in general.

Thanks.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: odd permutations form a group...identity

If you compose an odd permutation with an odd permutation, do you get an odd or even permutation?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: odd permutations form a group...identity

Can you prove that the identity is an even permutation? If you can, then you will also prove that it isn’t an odd permutation because no permutation can be both even an odd.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: odd permutations form a group...identity

(1) = (12)(12), an even permutation.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: odd permutations form a group...identity

I was replying to Clooney (only not as fast at typing as you).