What is the Cardinality of CG(K) in G mod Cg(K) when |K| = p and p is prime?

[chibulls59]

Homework Statement

Let G be a finite group and K a normal subgroup of G
If |K|=p where p is a prime
Prove that |G/C_G(K)| divides p-1

Homework Equations

The Attempt at a Solution

I must show that |G| / |C_G(K)| * something = p-1
I figured a good place to start would be to determine the cardinality of C_G(K) which I'm having trouble with.
So I just started writing down things that would help
Since K normal in G => N_G(K)=G
and since C_G(K) is normal in N_G(K) => C_G(K) is normal in G

 

[micromass]

Try to prove the following nice theorem:

N/C-theorem: If H is a subgroup of G, then N_G(H)/C_G(H) is isomorphic to a subgroup of Aut(G).

The proof of the theorem shouldn't be too difficult, as we can easily find a group homomorphism N_G(H)\rightarrow Aut(G) to which we can apply the first isomorphism theorem.

Now, the only thing you still need to show is that |Aut(\mathbb{Z}_p)|=p-1...

 

[chibulls59]

Since Aut(Z_p) is the set of maps that maps K-->K, wouldn't that mean that |Aut(Z_p)|=p?

 

[micromass]

Since Aut(Z_p) is the set of maps that maps K-->K, wouldn't that mean that |Aut(Z_p)|=p?

I don't immediately see why this would be the case. Can you give me p elements in Aut(Z_p) then?

Recall that elements in Aut(Z_p) are not simply maps Z_p--> Z_p, but this maps must actually be bijections!

Anyways, you could start by actually writing down some elements in Z_p and maybe noticing a general pattern...