Proving Alternating Groups in Sn: An Index 2 Subgroup

[zcdfhn]

Let A_n (the alternating group on n elements) consist of the set of all even permutations in S_n. Prove that A_n is indeed a subgroup of S_n and that it has index two in S_n and has order n!/2.

First of all, I need clarification on the definition of an alternating group. My book wasn't really good in explaining it.

My attempt at a solution is limited but I was thinking I can prove that A_n is a subgroup by showing that its elements are closed under composition and it is closed under inverses, but I need to better understand what an alternating group is before I start a proof. Also the order n!/2, I believe has to do with calculating permutations _nP_n-2 = n!/2! = n!/2.

Thanks in advance.

 

[Mark44]

Here's a Wikipedia article on alternating groups: http://en.wikipedia.org/wiki/Alternating_group

 

[sutupidmath]

A_n is simply the set of all even permutations. You know, i assume, what a permutation is right?

So, now you are asked to prove that actually the set of all even permutations, namely A_n, is a subgroup, when the opertaion is composition(multiplication). So all you need to show is that for any two elements of An, call them a,b, then ab is in A_n, that the inverse of every even permutaion is an even permutation, and you are done, siince there is a theorem that guarantees that any set that satisfies these conditions is a subgroup.

 

[zcdfhn]

thank you so much, it makes sense now

 

[sutupidmath]

Now the part of proving that the order of An is n!/2 requires more work.

First you probbably know that the order of Sn is n!. however we cannot rightaway conclude that An has n!/2 elements. We first need to show that there are as many even permutations as ther are odd permutations in Sn and then it is clear that the ord of An is n!/2.

Hint:

Let

$$S_n=(\alpha_1,\alpha_2,...,\alpha_k,\beta_1,\beta_2,...,\beta_r)$$

Be the set of all permutations where $$\alpha_i,\beta_j$$ are even and odd permutations respectively.

Now, what you can do is show that $$\beta_1\alpha_i$$ for i=1,..,k are all odd permutations...it requires a proof by contradiction somewhere along the lines.

Then again, you might want to show that $$\beta_1\beta_j$$ for j=1,...,r are all even permutations.

What do these two things tell you?