G-sets And Conjugacy

Question 1

What is a G-set?

Answer 1

Let G be a group and X a non-empty set, then Gnis said to act on X if there is a mapping,
* : G × X —> X
such that for all a, b, € G; x € X
1. (ab) * x = a * (b * x)
2. 1_G * x = x, where 1_G is the identity element of G.
Then X is said to be a G-set.

Question 2

What is a stabiliser (or isotropy group)?

Answer 2

Let x € X, where x is fixed and X is a G-set, then a stabiliser, Gx, is defined,
Gx : = {a € G: a * x = x}