Definition:Inner Automorphism
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Definition
Let $G$ be a group.
Let $x \in G$.
Let the mapping $\kappa_x: G \to G$ be defined such that:
- $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
$\kappa_x$ is called the inner automorphism of $G$ (given) by $x$.
The set of inner automorphisms of $G$ is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\Inn G$.
Also denoted as
While $\kappa$ is the symbol generally used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote an inner automorphism, this is not universal in the literature.
Different sources use different symbols, for example $\alpha$ as used by Allan Clark: Elements of Abstract Algebra.
The set of inner automorphisms of $G$ can be found denoted in a number of ways, for example:
- $\map {\mathscr I} G$
- $\map I G$
Also see
- Results about inner automorphisms can be found here.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.1$. Homomorphisms: Example $131$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.8$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{AA}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Proposition $8.17$