MaximalCondition 2003-10-04 11:41:34 2006-09-12 11:17:36 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A group is said to satisfy the \emph{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 \emph{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.