A group is said to satisfy the maximal condition if every strictly ascending chain of subgroups $$G_1 \subset G_2 \subset G_3 \subset \cdots$$ is finite.

This is also called the ascending chain condition.

A group satisfies the maximal condition if and only if the group and all its subgroups are finitely generated.

Similar properties are useful in other classes of algebraic structures: see for example the Noetherian condition for rings and modules.