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| Title of object: |
inner |
| Canonical Name: |
InnerAutomorphism |
| Type: |
Definition |
| Created on: |
2002-07-04 02:40:08.3218-04 |
| Modified on: |
2002-08-15 11:35:01.537714-04 |
| Classification: |
msc:20-00 |
| Defines: |
conjugation |
| Synonyms: |
inner=inner automorphism |
Preamble:
\usepackage{amsmath}
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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. 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. |
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