normal closure NormalClosure2 2004-10-06 05:55:33 2006-03-02 16:06:38 Definition nearly normal \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\normal{\trianglelefteq} Let $S$ be a subset of a group $G$. The \emph{normal closure} of $S$ in $G$ is the intersection of all normal subgroups of $G$ that contain $S$, that is \[\bigcap_{S\subseteq N\normal G}\!\!N.\] The normal closure of $S$ is the smallest normal subgroup of $G$ that contains $S$, and so is also called the \emph{normal subgroup generated by} $S$. It is not difficult to show that the normal closure of $S$ is the subgroup generated by all the conjugates of elements of $S$. The normal closure of $S$ in $G$ is variously denoted by $\langle S^G\rangle$ or $\langle S\rangle^G$ or $S^G$. If $H$ is a subgroup of $G$, and $H$ is of finite index in its normal closure, then $H$ is said to be \emph{nearly normal}.