SecondIsomorphismTheorem 2002-01-05 04:06:09 2007-07-04 05:42:00 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $(G,*)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$. Then \begin{itemize} \item $HK := \{ h*k \mid h \in H,\ k \in K \}$ is a subgroup of $G$, \item $K$ is a normal subgroup of $HK$, \item $H \cap K$ is a normal subgroup of $H$, \item There is a natural group isomorphism $H/(H \cap K) = HK/K$. \end{itemize} The same statement also holds in the category of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian category.