Is An×C2≆Sn?

manpreet Best Answer 2 years ago

I'm working on this exercise that appears in my group theory syllabus:

Prove An×C2≆SnAn×C2≇Sn for n≥3n≥3.

Since the chapter is about normal subgroups and factor groups, I might need to use that An⊲SnAn⊲Sn, but I don't see how to apply this. I found on the internet several proofs that the semi-direct product between AnAn and C2C2 is isomorphic to SnSn, but I have not yet seen semi-direct products and automorphism groups, so even if that applies here I'm looking for an answer not using this.

Actually I showed before the exact opposite, but there was a large mistake in my proof.

We assume n∈Zn∈Z such that n≥3n≥3. Let G1=AnG1=An and G2={(1),(1 2)}G2={(1),(1 2)}, and then define