Given a group G and a set X, if the action of an element g of G leaves an element x of X fixed, we call g a stabilizer of the point x. The set of all stabilizers of a point x form, in fact, a subgroup. This subgroup is called the stabilizer group of x. If every stabilizer group is trivial, trivial meaning contains only the identity element of G, then we say that the action of G on the set X is “free.” There is no an and b combination, where a is an element of the group G and b is an element of the set X that leaves b unmoved, unless a is the identity element of G.
We say that the action of G on X is free. Free action or free group action or acts freely.
If the action of a group G on a set X is free, then, if you take any element of G (not the identity though) and act on any element x of X, then you will get a distinct element. Distinct means that two different elements will not send x to the same element y in X.
For example, if there are 20 elements in a group G and 500 elements in a set X, if you apply all 20 elements of G to an element x of X, you will get back 20 elements where 1 is x (due to the identity) and the other 19 are distinct.
This is actually used in constructing manifolds and, if you have orbits, which we will discuss next, orbifolds. They are also used in constructing principal G-bundles, which is all the rage nowadays.
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