# kenrnal of a map for group action

• Nov 25th 2013, 07:06 AM
topsquark
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
• Nov 25th 2013, 06:28 PM
Haven
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 $\displaystyle g\in G$
• Nov 26th 2013, 01:37 AM
topsquark
Re: kenrnal of a map for group action
Quote:

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