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| Title of object: |
opposite group |
| Canonical Name: |
OppositeGroup |
| Type: |
Definition |
| Created on: |
2007-05-27 21:05:28 |
| Modified on: |
2007-05-27 21:43:16 |
| Classification: |
msc:20-00, msc:08A99 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
\usepackage{psfrag}
\usepackage{graphicx}
\usepackage{amsthm}
\usepackage{xypic}
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Content:
Let $G$ be a group under the operation $*$. The \emph{opposite group} of $G$, denoted $G^{\mathrm{op}}$, has the same underlying set as $G$, and its group operation is $*'$ defined by $g_1*'g_2=g_2*g_1$.
If $G$ is abelian, then it is equal to its own opposite group. On the other hand, a nonabelian group $G$ can be isomorphic to its own opposite group. For example, since ${S_3}^{\mathrm{op}}$ is a nonabelian group of order six, $S_3 \cong {S_3}^{\mathrm{op}}$.
\PMlinkescapetext{Similar} constructions occur in opposite ring and opposite category.
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