| Version 15 |
Version 14 |
| \PMlinkescapeword{entire} |
\PMlinkescapeword{entire} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
|
|
| The \emph{center} of a group $G$ is the subgroup consisting of those elements that commute with every other element. Formally, |
The \emph{center} of a group $G$ is the subgroup consisting of those elements that commute with every other element. Formally, |
| $$\operatorname{Z}(G) = \{x \in G \mid xg = gx\hbox{ for all }g \in G\}.$$ |
$$\operatorname{Z}(G) = \{x \in G \mid xg = gx\hbox{ for all }g \in G\}.$$ |
|
|
| It can be shown that the center has the following properties: |
It can be shown that the center has the following properties: |
| \begin{itemize} |
\begin{itemize} |
| \item It is a normal subgroup (in fact, a characteristic subgroup). |
\item It is a normal subgroup (in fact, a characteristic subgroup). |
| \item It consists of those conjugacy classes containing just one element. |
\item It consists of those conjugacy classes containing just one element. |
| \item The center of an abelian group is the entire group. |
\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. |
\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}) |
(\PMlinkname{Proof of a stronger version of this theorem.}{ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection}) |
| \end{itemize} |
\end{itemize} |
|
|
| A subgroup of the center of a group $G$ |
|
| is called a {\emph central subgroup} of $G$. |
|
|
|
| For any group $G$, the \PMlinkname{quotient}{QuotientGroup} $G/\operatorname{Z}(G)$ is called the \emph{central quotient} of $G$, |
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 $\Inn(G)$. |
and is isomorphic to the inner automorphism group $\Inn(G)$. |
