center of a group GroupCentre 2002-02-19 11:40:35 2008-10-16 14:48:34 Definition central quotient \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \DeclareMathOperator{\Inn}{Inn} \PMlinkescapeword{entire} \PMlinkescapeword{properties} The \emph{center} of a group $G$ is the subgroup consisting of those elements that commute with every other element. Formally, $$\operatorname{Z}(G) = \{x \in G \mid xg = gx\hbox{ for all }g \in G\}.$$ It can be shown that the center has the following properties: \begin{itemize} \item It is a normal subgroup (in fact, a characteristic subgroup). \item It consists of those conjugacy classes containing just one element. \item The center of an abelian group is the entire group. \item For every prime $p$, every non-trivial finite \PMlinkname{$p$-group}{PGroup4} has a non-trivial center. (\PMlinkname{Proof of a stronger version of this theorem.}{ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection}) \end{itemize} A subgroup of the center of a group $G$ is called a \emph{central subgroup} of $G$. All central subgroups of $G$ are normal in $G$. For any group $G$, the \PMlinkname{quotient}{QuotientGroup} $G/\operatorname{Z}(G)$ is called the \emph{central quotient} of $G$, and is isomorphic to the inner automorphism group $\Inn(G)$.