Definition:Inner Automorphism

From ProofWiki
Jump to navigation Jump to search


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.