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 $\forall g \in G: \kappa_x \left({g}\right) = x g x^{-1}$.

$\kappa_x$ is called the inner automorphism of $G$ given by $x$.

The set of all inner automorphisms of $G$ is denoted $\operatorname{Inn} \left({G}\right)$ or $\mathscr I \left({G}\right)$.

Also see

• Inner Automorphism is Automorphism

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 7.1$: Example $131$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $11.8 \ \text{(a)}$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Problem $\text{AA}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 64$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Proposition $8.17$