 Maximal condition on subgroups

A group has the maximal condition on subgroups when every ascending sequence of subgroups eventually stabilizes. Equivalently, every subgroup is a finitely generated group. If a subgroup were not finitely generated, repeatedly adjoining an element outside the previously generated subgroup would give a strictly ascending sequence. Conversely, generators of the union of an ascending sequence all lie in one term, making that term the whole union. This condition is preserved by group extensions because each subgroup is an extension of its intersection with the kernel by its image in the quotient.