Power of Group Element in Kernel of Homomorphism iff Power of Image is Identity

Theorem

Let $G$ be a group whose identity is $e_G$.

Let $H$ be a group whose identity is $e_H$.

Let $\phi: G \to H$ be a (group) homomorphism.

Let $x^n \in \map \ker \phi$ for some integer $n$.

Then:

$\paren {\map \phi x}^n = e_H$

Proof

\(\ds x^n\)

\(\in\)

\(\ds \map \ker \phi\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \map \phi {x^n}\)

\(=\)

\(\ds e_H\)

Definition of Kernel of Group Homomorphism

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {\map \phi x}^n\)

\(=\)

\(\ds e_H\)

Homomorphism of Power of Group Element

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $8$