Dummit and Foote Section 1.7 Group Actions - Exercise 4 is as follows:

Let G be a group acting on a set A and fix some a A.

Show that the following sets are subgroups of G:

(a) the kernel of the action

(b) {g G | ga = a} {stabilizer of a in G}

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I am working on part (a) and have not got far!

We know that the kernel of the action of G on A is as follows:

Kernel, K = {g G| ga = a for all a A}

We need to show x K for all x, y K

So assume , K

Then a = a and a = a

Now need to show K

That is ( )a = a

Now ( )a = ( a)

But ??? where to from here

Can anyone help? Be grateful for some assistance!

Peter