# Given a subgroup A in G, consider the normalizer. Prove A is normal iff normalizer=G.

• Mar 2nd 2011, 07:24 PM
JJMC89
Given a subgroup A in G, consider the normalizer. Prove A is normal iff normalizer=G.
Given a subgroup $\displaystyle A \subseteq G$, consider the set $\displaystyle N_G(A) = \{g \in G | gAg^{-1} = A\}$.

Prove that $\displaystyle A$ is normal if and only if $\displaystyle N_G(A) = G$.
• Mar 2nd 2011, 07:32 PM
Drexel28
Quote:

Originally Posted by JJMC89
Given a subgroup $\displaystyle A \subseteq G$, consider the set $\displaystyle N_G(A) = \{g \in G | gAg^{-1} = A\}$.

Prove that $\displaystyle A$ is normal if and only if $\displaystyle N_G(A) = G$.

This is by definition isn't it......namely $\displaystyle N_G(A)$ is the largest subgroup $\displaystyle H$ (and yes, it is always a subgroup) such that $\displaystyle A\unlhd H$
• Mar 2nd 2011, 07:36 PM
JJMC89
Quote:

Originally Posted by Drexel28
This is by definition isn't it......namely $\displaystyle N_G(A)$ is the largest subgroup $\displaystyle H$ (and yes, it is always a subgroup) such that $\displaystyle A\unlhd H$

The only definition I have for the normalizer is that it is the set $\displaystyle N_G(A) = \{g \in G | gAg^{-1} = A\}$.
• Mar 2nd 2011, 07:54 PM
Drexel28
Quote:

Originally Posted by JJMC89
The only definition I have for the normalizer is that it is the set $\displaystyle N_G(A) = \{g \in G | gAg^{-1} = A\}$.

I mean for normality. One has that $\displaystyle \displaystyle N\unlhd G$ if and only if $\displaystyle gNg^{-1}=N$ for all $\displaystyle g\in G$ which is precisely saying that $\displaystyle N_g(N)=N$.
• Mar 2nd 2011, 08:01 PM
JJMC89
Quote:

Originally Posted by Drexel28
I mean for normality. One has that $\displaystyle \displaystyle N\unlhd G$ if and only if $\displaystyle gNg^{-1}=N$ for all $\displaystyle g\in G$ which is precisely saying that $\displaystyle N_g(N)=N$.