Composite of Group Homomorphisms is Homomorphism/Proof 2
Theorem
Let:
- $\struct {G_1, \circ}$
- $\struct {G_2, *}$
- $\struct {G_3, \oplus}$
be groups.
Let:
- $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$
- $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$
be homomorphisms.
Then the composite of $\phi$ and $\psi$ is also a homomorphism.
Proof
So as to alleviate possible confusion over notation, let the composite of $\phi$ and $\psi$ be denoted $\psi \bullet \phi$ instead of the more usual $\psi \circ \phi$.
Then what we are trying to prove is denoted:
- $\paren {\psi \bullet \phi}: \struct {G_1, \circ} \to \struct {G_3, \oplus}$ is a homomorphism.
To prove the above is the case, we need to demonstrate that the morphism property is held by $\circ$ under $\psi \bullet \phi$.
We take two elements $x, y \in G_1$, and put them through the following wringer:
\(\ds \map {\paren {\psi \bullet \phi} } {x \circ y}\) | \(=\) | \(\ds \map \psi {\map \phi {x \circ y} }\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \psi {\map \phi x * \map \phi y}\) | Definition of Morphism Property: applied to $\circ$ under $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \psi {\map \phi x} \oplus \map \psi {\map \phi y}\) | Definition of Morphism Property: applied to $*$ under $\psi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {\psi \bullet \phi} } x \oplus \map {\paren {\psi \bullet \phi} } y\) | Definition of Composition of Mappings |
Disentangling the confusing and tortuous expressions above, we (eventually) see that this shows that the morphism property is indeed held by $\circ$ under $\psi \bullet \phi$.
Thus $\paren {\psi \bullet \phi}: \struct {G_1, \circ} \to \struct {G_3, \oplus}$ is indeed a homomorphism.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.2$. Some lemmas on homomorphisms: Lemma $\text{(iv)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 60$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $3 \ \text{(i)}$