Definition:Inner Automorphism

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

• Inner Automorphism is Automorphism

• Inner Automorphisms form Normal Subgroup of Automorphism Group

• Definition:Inner Automorphism Group

• 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 $8.17$