Prove that if and are normal subsets of a Sylow -subgroup of a finite group then is -conjugate to if and only if is -conjugate to . Deduce that two elements in the center of are conjugate in if and only if they are conjugate in . (A subset is normal in if .)

Let with and .

The direction is immediate. To see , suppose is -conjugate to ; then there exists such that and . Note that . Now . (@@@)

Now suppose . In this case, and , so that and are normal in . By the above argument we have that and are -conjugate if and only if they are -conjugate.

## Comments

In the second paragraph, you said I can see that which is all you need for this proof. But how did you get the other inclusion?

Good question. I’m not convinced now that that is true.

I’ve got another proof that appears to work, though.

Thanks!

I have a problem showing is clear, but for the new proof you need It’s the same problem I had with showing the other inclusion in my previous comment.

Your original proof didn’t work because the equalities might not have been true. But, the supersets were obvious, and they were sufficient. Here’s the same proof with equalities replaced by supersets.

Let Then so because P is sylow p. Therefore U and W are conjugate in

Aha- my proof only shows that , from which the inequality follows but not equality. That’s what I get for trying to think at 6 AM.đź™‚

Thanks!

Okay… now I’m not so convinced the proof works. I don’t have the time at the moment to dig into it, so I’ll mark this as ‘incomplete’ and come back to it later.

Ah, typo. The second last term in the long chain should have been They really should let you preview comments.