generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group.

Generating set of a group: A set of group elements which are not contained in any subgroup of the group other than the entire group itself.

Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

The image under a Nielsen transformation (elementary or not, regular or not) of a generating set of a group G is also a generating set of G. Two generating sets are called Nielsen equivalent if there is a Nielsen transformation taking one to the other.