This can be understood as an example of the group action of 'G' on the elements of 'G'.

Each word in S represents an element of G, namely the product of the expression.

(For example, one element of G is the simultaneous translation of all particles and fields forward in time by five seconds.)

The only group axiom that requires some effort to verify is that each element of G is invertible.

Therefore every element of G is invertible, so as remarked earlier, G is a group.

Let "x" be an element of "G" not in "E".

In this case "g" is called an "elliptic" element of "G".

Conjugating by an element of "G", the smaller disk can be taken to have centre 0.

To save words, they say "a is in G" to mean "a is an element of G".

When restricted to the unipotent elements of G it becomes the Springer resolution.