Prove that contains a subgroup isomorphic to for all .

Let . We saw in a previous theorem that where .

Let , and let . Define by if is even and if is odd.

- is a homomorphism: If and are even, then . If without loss of generality is even and is odd, . If and are odd, then .
- is injective: Suppose . Then since is even and , we have . So the kernel of is trivial and is injective.

So embeds as a subgroup of . Moreover, recall that where . So this copy of in is generated by .

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## Comments

Hi, just wondering why is step 3 necessary.

If you have your injective isomorphism, then phi(Sn) is a subgroup of A_n isomorphic to S_n-2, right?

Step 3 is indeed superfluous. At first I thought it was useful for giving a specific generating set for the copy of in , but once we know that is injective, we get the generating set for free.

Thanks!

Quick question: why can we commute \sigma and \beta when beta is odd?

They are disjoint. That is, fixes the elements moves and vice versa. It’s not too hard to show that disjoint permutations commute. (D&F discuss this on page 32.)