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'center of a group'
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| Title of object: |
center of a group |
| Canonical Name: |
GroupCentre |
| Type: |
Definition |
| Created on: |
2002-02-19 11:40:35 |
| Modified on: |
2008-04-27 17:24:41 |
| Classification: |
msc:20A05 |
| Defines: |
central quotient |
| Synonyms: |
center of a group=center
center of a group=centre |
Revision comment (for changes between this and next version):
| all central subgroups of G are normal in G |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\DeclareMathOperator{\Inn}{Inn} |
Content:
\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 $G$.
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 $\Inn(G)$. |
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