fully invariant subgroup FullyInvariantSubgroup 2002-12-07 20:46:22 2004-03-22 16:31:44 Definition fully invariant fully characteristic fully characteristic subgroup % 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 subgroup $H$ of a group $G$ is \emph{fully invariant} if $f(H) \subseteq H$ for all endomorphisms $f \colon G \to G$. Such a subgroup is also called \emph{fully characteristic}. This is a stronger condition than being a characteristic subgroup. The derived subgroup is fully invariant.