Kernel is Trivial iff Monomorphism/Group
Theorem
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
Then $\phi$ is a group monomorphism if and only if $\map \ker \phi$ is trivial.
Proof
Necessary Condition
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group monomorphism.
By Homomorphism to Group Preserves Identity, $e_S \in \map \ker \phi$.
If $\map \ker \phi$ contained another element $s \ne e_S$, then $\map \phi s = \map \phi {e_S} = e_T$ and $\phi$ would not be injective, thus not be a group monomorphism.
So $\map \ker \phi$ can contain only one element, and that must be $e_S$, which is therefore the trivial subgroup of $S$.
$\Box$
Sufficient Condition
Now suppose $\map \ker \phi = \set {e_S}$.
Then, for any $x, y \in S$:
\(\ds \map \phi x\) | \(=\) | \(\ds \map \phi y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | \(=\) | \(\ds \map \phi y * \paren {\map \phi y}^{-1}\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {x \circ y^{-1} }\) | \(=\) | \(\ds e_T\) | Definition of Morphism Property | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \circ y^{-1}\) | \(\in\) | \(\ds \map \ker \phi\) | Definition of Kernel of Group Homomorphism | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \circ y^{-1}\) | \(=\) | \(\ds e_S\) | by hypothesis | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds e_S \circ y\) | Group Axiom $\text G 2$: Existence of Identity Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds y\) | Definition of Identity Element |
Thus $\phi$ is injective, and therefore a group monomorphism.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: Example $139$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $22$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Quotient Groups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.8$ Homomorphisms and their elementary properties
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Exercise $1$
- 2002: John B. Fraleigh: A First Course in Abstract Algebra (7th ed.): Chapter $13$: Corollary $13.18$