Why is the number of fixed points divisible by p in Cauchy's Theorem proof?

[Bleys]

I'm going through the proof of Cauchy's Theorem in a textbook and I'm stuck on a particular point. I've tried looking at other proofs, but they all use the Orbit-Stabilizer Theorem or induction. The proof presented in the book probably uses the O-S Theorem, but not directly (since it's not been covered yet).

Let G be a finite group and p a prime dividing the order of G. Define the set A= \{ (g_{1},...,g_{p} ) : g_{i} \in G , g_{1}...g_{p} = 1 \} and a permutation \pi on A by \pi (g_{1},...,g_{p}) = (g_{2},...,g_{p},g_{1})
It shows that if g^p=1, then (g,...,g) is in A and is fixed by \pi, and that all other elements of A belong to cycles of size p. That's all fine. Then it shows |A| = |G|^{p-1} and so p divides |A|. Since all cycles have size 1 or p, the number of fixed points is also divisible by p. I don't understand this. Why does the fact that |A| = pk imply that the number of fixed points is also a multiple of p?

All I've been able to deduce is that kp = |A| = an_{1} + bn_{2} where n_{1}, n_{2} is the number of 1 cycles and p cycles, respectively.
Could someone clear this up for me, please?

(also in LaTeX, why are \pi and n_{1},n_{2} not aligned with the rest of the line, and the curly brackets not showing up in the code for the definition of A? EDIT: fixed now, thank you Gregg!)

 

[Gregg]

(also in LaTeX, why are \pi and n_{1},n_{2} not aligned with the rest of the line, and the curly brackets not showing up in the code for the definition of A?)

use \{

A = \{ x : x^2=1\}

second, use itex tags instead of tex:

I like to talk about the number\pi during a sentence.

 

[Martin Rattigan]

...Then it shows |A| = |G|^{p-1} and so p divides |A|. Since all cycles have size 1 or p, the number of fixed points is also divisible by p. I don't understand this. Why does the fact that |A| = pk imply that the number of fixed points is also a multiple of p?
...

The sequences g₁,g₂,...,g_p (if any) where g₁g₂...g_p=e and it isn't the case that g₁=g₂=...=g_p fall into sets of p that are cyclic permutations of each other, because if g₁g₂...g_p=e then g₂g₃...g_pg₁=e etc.

If we remove these from A, since |A| = pk that leaves some multiple m=rp of p sequences ggg...g of elements such that g^p=e. One of these is eee...e, so r is not zero. Therefore there must be rp-1>0 elements of order p.

(Incidentally these fall into subgroups of order p that intersect only in the identity and each contains p-1 elements of order p, so you can also say rp-1=0 (p-1) or r=1 (p-1), so that the number of elements of order p is actually
(h(p-1)+1)p-1=(hp+1)(p-1) for some integer h.)

By the way, I believe this is McKay's proof of Cauchy's theorem , first published in American Mathematical Monthly, Vol.66 (1959), page 119.

 

[Bleys]

...fall into sets of p that are cyclic permutations of each other...

I'm assuming you mean sets of order p. Ok, it was not obvious to me that the set of non-fixed was also divisible by p (but I see how the result follows from this).
Thanks for the help (and the reference to McKay)!

 

[daveyp225]

The sequences g₁,g₂,...,g_p (if any) where g₁g₂...g_p=e and it isn't the case that g₁=g₂=...=g_p fall into sets of p that are cyclic permutations of each other, because if g₁g₂...g_p=e then g₂g₃...g_pg₁=e etc.

If we remove these from A, since |A| = pk that leaves some multiple m=rp of p sequences ggg...g of elements such that g^p=e. One of these is eee...e, so r is not zero. Therefore there must be rp-1>0 elements of order p.

(Incidentally these fall into subgroups of order p that intersect only in the identity and each contains p-1 elements of order p, so you can also say rp-1=0 (p-1) or r=1 (p-1), so that the number of elements of order p is actually
(h(p-1)+1)p-1=(hp+1)(p-1) for some integer h.)

By the way, I believe this is McKay's proof of Cauchy's theorem , first published in American Mathematical Monthly, Vol.66 (1959), page 119.

Found this thread in google (yes, old, i know), and I'm having trouble accepting this as a trivial statement (bold). Is there an easy way to just "see" this? I believe its truth rests in the fact p is prime, but it wasn't mentioned here...