Kernel of Group Homomorphism is Subgroup

Theorem

The kernel of a group homomorphism is a subgroup of its domain:

$\map \ker \phi \le \Dom \phi$

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.

From Homomorphism to Group Preserves Identity, $\map \phi {e_G} = e_H$, so $e_G \in \map \ker \phi$.

Therefore $\map \ker \phi \ne \O$.

Let $x, y \in \map \ker \phi$, so that $\map \phi x = \map \phi y = e_H$.

Then:

$\ds \map \phi {x^{-1} \circ y}$

$=$

$\ds \map \phi {x^{-1} } * \map \phi y$

Definition of Morphism Property

$\ds $

$=$

$\ds \paren {\map \phi x}^{-1} * \map \phi y$

$\ds $

$=$

$\ds e_H^{-1} * e_H$

as $x, y \in \map \ker \phi$

$\ds $

$=$

$\ds e_H$

Definition of Identity Element

So $x^{-1} \circ y \in \map \ker \phi$, and from the One-Step Subgroup Test, $\map \ker \phi \le \Dom \phi$.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image
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: Morphisms
1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.7$ Homomorphisms and their elementary properties