minimal condition

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

$$G_1\supset G_2\supset G_3\supset \mathrm{\cdots }$$

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

$$⟨x⟩\supset ⟨x^2⟩\supset ⟨x^4⟩\supset \mathrm{\cdots }\supset ⟨x^{2^n}⟩\supset \mathrm{\cdots }$$

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.