Group Homomorphism Preserves Identity/Proof 1
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Theorem
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.
Let:
Then:
- $\map \phi {e_G} = e_H$
Proof
\(\ds \map \phi {e_G}\) | \(=\) | \(\ds \map \phi {e_G \circ e_G}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {e_G} * \map \phi {e_G}\) | Definition of Morphism Property |
That is:
\(\ds \map \phi {e_G} * e_H\) | \(=\) | \(\ds \map \phi {e_G} * \map \phi {e_G}\) | Definition of Identity Element | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds e_H\) | \(=\) | \(\ds \map \phi {e_G}\) | Cancellation Laws |
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.1$ Homomorphisms and their elementary properties
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms