Thread: Algebra, group action

Studying for a final and had the following practice problem, which I can't answer. Could anyone help?

Let G be a finite group acting on a finite set. Prove that <a>=<b> for two elements a,b in G then I(a)=I(b), where for any element g in G, I(g)={s in S: gs=s}.

Thank you.

Originally Posted by PvtBillPilgrim

Studying for a final and had the following practice problem, which I can't answer. Could anyone help?

Let G be a finite group acting on a finite set. Prove that <a>=<b> for two elements a,b in G then I(a)=I(b), where for any element g in G, I(g)={s in S: gs=s}.

Thank you.

I presume you are talking about G-sets.
The notation I will use is, for the group.
And for the non-empty set.
The binary operation will simply be juxtaposition of two elements,

The standard way to show,

It to show,

Now,



Let,

Then,

Because,

Thus,
implies
Similarly,

Thus,
implies .
Thus,


(The facts that the sets are finites and groups are finite was not necessary).