# Kernel of a group action and Sa

• Oct 19th 2011, 04:48 AM
Bernhard
Kernel of a group action and Sa
Dummit and Foote Section 1.7 Exercise 5 reads as follows:

Prove that the kernel of an action of the group G on a set A is the same as the kernel of the corresponding permutation representation G $\longrightarrow$ $S_A$

I am having real trouble with this - can anyone help me?

By the way - what is the meaning of "the corresponding permutation representation $\longrightarrow$ $S_A$" ???

I would be really grateful for some assistance.

Peter
• Oct 19th 2011, 07:02 AM
Deveno
Re: Kernel of a group action and Sa
there are two ways to look at a group action.

1)as a "multiplication" GxA-->A given by (g,a)-->g.a (also written as g(a)).

2) as a homomorphism φ from G--->Sym(A) (the set of bijections on A, so each element of g maps to a permutation $\sigma_g$ of A).

now, the kernel as per the first way of looking at a group action is the set {g in G: g.a = a, for all a in A}.

the kernel in the second way of looking at a group action is the set {g in G : $\sigma_g = id_A$, the identity function on A}

we want to show these are the same thing. so suppose g.a = a, for all a in A. then
$\sigma_g(a) = a$ for all a in A, that is $\phi(g) = \sigma_g = id_A$, so $g \in ker(\phi)$.

on the other hand, suppose $g \in ker(\phi)$. then g.a = $\sigma_g(a) = \phi(g)(a) = id_A(a)$ = a, so that g is in the set {g in G: g.a = a, for all a in A}.

the whole point of this exercise is to show you the two definitions of a group action "really are the same".