Stabilizers

• May 21st 2008, 08:05 AM
topsquark
Stabilizers
Okay, just to make sure I have this straight:

Given a group G and an element g of G

The "stabilizer" of g is the subset $\displaystyle \{x \} \subset G$ such that $\displaystyle yg = g$ for all $\displaystyle y \in \{ x \}$.

And the stabilizer of g is a subgroup of G, right?

Thanks!
-Dan
• May 21st 2008, 08:10 AM
ThePerfectHacker
Quote:

Originally Posted by topsquark
Okay, just to make sure I have this straight:

Given a group G and an element g of G

The "stabilizer" of g is the subset $\displaystyle \{x \} \subset G$ such that $\displaystyle yg = g$ for all $\displaystyle y \in \{ x \}$.

And the stabilizer of g is a subgroup of G, right?

Thanks!
-Dan

You got the elements of the group confused with elements of the set.

Let $\displaystyle G$ be a a group, and let $\displaystyle X$ be a set. Let $\displaystyle G$ act on $\displaystyle X$.
Given an element $\displaystyle x\in X$ the stabalizer, $\displaystyle G_x$, is the set $\displaystyle \{g\in G|gx=x\}$. And yes, the stabalizer is a subgroup.

I can give some examples if you want to.
• May 21st 2008, 08:12 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
You got the elements of the group confused with elements of the set.

Let $\displaystyle G$ be a a group, and let $\displaystyle X$ be a set. Let $\displaystyle G$ act on $\displaystyle X$.
Given an element $\displaystyle x\in X$ the stabalizer, $\displaystyle G_x$, is the set $\displaystyle \{g\in G|gx=x\}$. And yes, the stabalizer is a subgroup.

I can give some examples if you want to.

Ahhh! The lightbulb just went on. I think I've got it now. I'll play around with it a bit and let you know if I need more. Thanks!

-Dan