minimal condition

A group is said to satisfy the minimal condition if every strictly descending chain of subgroups

G_{1}\supset G_{2}\supset G_{3}\supset\cdots

is finite.

This is also called the descending chain condition.

A group which satisfies the minimal condition is necessarily periodic. For if it contained an element $x$ of infinite order, then

\langle x\rangle\supset\langle x^{2}\rangle\supset\langle x^{4}\rangle\supset% \cdots\supset\langle x^{2^{n}}\rangle\supset\cdots

is an infinite descending chain of subgroups.

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