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'center'
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| Title of object: |
center |
| Canonical Name: |
GroupCentre |
| Type: |
Definition |
| Created on: |
2002-02-19 11:40:35-05 |
| Modified on: |
2002-04-15 16:16:48-04 |
| Classification: |
msc:20A05 |
| Synonyms: |
center=centre |
Revision comment (for changes between this and next version):
| Changes for correction #1562 ('Change title to center (group)'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
The center of a group $G$ is the subgroup of elements which commute with every other element. Formally
$$\operatorname{Z}(G) = \{x \in G \mid xg = gx,\ \forall \ g \in G\}$$
It can be shown that the center has the following properties
\begin{itemize}
\item It is non-empty since it contains at least the identity element
\item It consists of those conjugacy classes containing just one element
\item The center of an abelian group is the entire group
\item It is normal in $G$
\item Every p-group has a non-trivial center
\end{itemize} |
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