# Thread: kenrnal of a map for group action

1. ## kenrnal of a map for group action

Okay, given a function for a group action of the group G acting on a set A (with a fixed point a of A): $\displaystyle \phi _a : G \times \{ a \} \to A$. One definition I have for the kernal is $\displaystyle ker \phi _a = \{ (g, a) | \phi _a (g, a) = 1_A ~ \forall g \in G \}$. But the definition in the group action section is that $\displaystyle ker \phi _a = \{ g \in G | g \cdot a = a \}$. Are these somehow equivalent? I don't see it.

-Dan

2. ## Re: kenrnal of a map for group action

I don't understand the definition of $\displaystyle \phi_a$. If a is a fixed point of G, then g.a=a for all g. So what is $\displaystyle phi_a(g,a)$ for $\displaystyle g\in G$

3. ## Re: kenrnal of a map for group action

Originally Posted by Haven
I don't understand the definition of $\displaystyle \phi_a$. If a is a fixed point of G, then g.a=a for all g. So what is $\displaystyle phi_a(g,a)$ for $\displaystyle g\in G$
In general the group action of a group G acting on an element of a set A is defined as $\displaystyle \phi : G \times A \to A :g \cdot a \mapsto a'$. The group G essentially acts as a permutation on the set A. If we restrict ourselves to a fixed element a of A (not G) then we have $\displaystyle \phi _a : G \times \{ a \} \to A$. My question is how to define the kernal of $\displaystyle \phi _a$.

-Dan