1. ## Simple question about a definition

I have this in my lecture notes:

Let G be a group and let P and Q be p-subgroups. Suppose that Q normalizes
P. Then PQ is also a p-subgroup.

However it doesnt say anywhere what normalizes means, I know what the normaliser is

$\displaystyle N_G (Q) = \{ g \in G : gQ = Qg \}$, $\displaystyle N_G (P) = \{ g \in G : gP = Pg \}$.

I was wondering if anyone could tell me what Q normalises P means? Thanks very much

2. Originally Posted by slevvio
I have this in my lecture notes:

Let G be a group and let P and Q be p-subgroups. Suppose that Q normalizes
P. Then PQ is also a p-subgroup.

However it doesnt say anywhere what normalizes means, I know what the normaliser is

$\displaystyle N_G (Q) = \{ g \in G : gQ = Qg \}$, $\displaystyle N_G (P) = \{ g \in G : gP = Pg \}$.

I was wondering if anyone could tell me what Q normalises P means? Thanks very much

$\displaystyle Q$ normalizes $\displaystyle P\iff Q\subset N_G(P)\iff P^q=q^{-1}Pq=P\,\,\,,\,\forall\,q\in Q$

Tonio