Homomorphism to Group Preserves Inverses

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {T, *}$ be a group.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Let $x^{-1}$ be an inverse of $x$ for $\circ$.


Then $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.


Proof

By hypothesis, $\struct {T, *}$ is a group.

By Group Axiom $\text G 2$: Existence of Identity Element, $\struct {T, *}$ has an identity.

Thus Homomorphism with Identity Preserves Inverses can be applied.

$\blacksquare$


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.2$. Some lemmas on homomorphisms: Lemma $\text{(iii)}$
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.3: \ 3^\circ$
  • 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $22$
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