Property of Group Automorphism which Fixes Identity Only/Corollary 2

Corollary to Property of Group Automorphism which Fixes Identity Only

Let $G$ be a finite group whose identity is $e$.

Let $\phi: G \to G$ be a group automorphism.

Let $\phi$ have the property that:

$\forall g \in G \setminus \set e: \map \phi t \ne t$

That is, the only fixed element of $\phi$ is $e$.

Let:

$\phi^2 = I_G$

where $I_G$ denotes the identity mapping on $G$.

Then:

$\forall g \in G: \map \phi g = g^{-1}$

Proof

Let $g \in G$.

Then:

\(\ds \exists x \in G: \, \)

\(\ds \map \phi g\)

\(=\)

\(\ds \map \phi {x^{-1} \, \map \phi x}\)

Corollary 1

\(\ds \)

\(=\)

\(\ds \paren {\map \phi x}^{-1} x\)

\(\ds \)

\(=\)

\(\ds g^{-1}\)

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $26$