|
|
|
Viewing Version
3
of
'inner'
|
[ view 'inner'
|
back to history
]
| Title of object: |
inner |
| Canonical Name: |
InnerAutomorphism |
| Type: |
Definition |
| Created on: |
2002-07-04 02:40:08.3218-04 |
| Modified on: |
2002-08-16 09:53:30.285691-04 |
| Classification: |
msc:20-00 |
| Defines: |
conjugation, outer, outer automorphism |
| Synonyms: |
inner=inner automorphism |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{theorem}[proposition]{Theorem} |
Content:
Let $G$ be a group. For every $x\in G$, we define a
mapping
$$\phi_x:G\rightarrow G,\quad y\mapsto x y x^{-1},\quad y\in G,$$
called conjugation by $x$.
It is easy to show the conjugation map is in fact, a group automorphism.
An automorphism of $G$ that corresponds to the conjugation by some
$x\in G$ is called inner. An automorphism that isn't inner is called
an \emph{outer} automorphism. The set of all inner automorphisms of
$G$ forms a group, often denoted as $\mathop{\mathrm{Inn}}(G)$, which
is isomorphic to the quotient of $G$ by $Z(G)$, the centre subgroup. |
|
|
|
|
|
