Generating Sets in Group Theory

In mathematics, a generating set (or generating family) of a group is a subset of the group whose elements can be combined to produce all the other elements of the group. In other words, if we take any element of the group, we can write it as a product of elements of the generating set.

For example, the set {1, 2, 3} is a generating set of the group of integers under addition, because we can write any integer as a sum of these elements:

1 + 2 + 3 = 6
2 + 3 + 1 = 6
3 + 1 + 2 = 6

In this case, the generating set consists of three elements, but there are many other possible generating sets for the same group. For example, {2, 3, 4} is also a generating set of the group of integers under addition, because we can write any integer as a sum of these elements:

2 + 3 + 4 = 9
3 + 4 + 2 = 9
4 + 2 + 3 = 9

In general, a generating set need not consist of all the elements of the group, and it may be easier to work with a smaller generating set than with the entire group.

The term "generatrices" is not commonly used in mathematics, but it appears that you are using it to refer to the elements of a generating set. In this case, the generatrices would be the elements {1, 2, 3} or {2, 3, 4}, depending on which generating set we choose to use.