In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.[1]
By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
- ^ Gregorac, Robert J. (1967). "A note on finitely generated groups". Proceedings of the American Mathematical Society. 18 (4): 756–758. doi:10.1090/S0002-9939-1967-0215904-3.