Group Homomorphism of Product with Inverse
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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.
Then:
\(\text {(1)}: \quad\) | \(\ds \forall x, y \in G: \, \) | \(\ds \map \phi {x \circ y^{-1} }\) | \(=\) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \forall x, y \in G: \, \) | \(\ds \map \phi {y^{-1} \circ x}\) | \(=\) | \(\ds \paren {\map \phi y}^{-1} * \map \phi x\) |
where $y^{-1}$ denotes the inverse of $y$.
Proof
Let $e_G$ and $e_H$ be the identities of $\struct {G, \circ}$ and $\struct {H, *}$ respectively.
By Group Axiom $\text G 0$: Closure:
- $\forall x, y \in G: x \circ y^{-1} \in G, y^{-1} \circ x \in G$
- Result $(1)
- $:
\(\ds \map \phi {x \circ y^{-1} } * \map \phi y\) | \(=\) | \(\ds \map \phi { \paren {x \circ y^{-1} } \circ y}\) | Definition of Group Homomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {x \circ \paren { y^{-1} \circ y} }\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {x \circ e_G}\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x\) | Definition of Identity Element | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\map \phi {x \circ y^{-1} } * \map \phi y} * \paren {\map \phi y}^{-1}\) | \(=\) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \map \phi {x \circ y^{-1} } * \paren {\map \phi y * \paren {\map \phi y}^{-1} }\) | \(=\) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \map \phi {x \circ y^{-1} } * e_H\) | \(=\) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | Definition of Inverse Element | |||||||||||
\(\ds \map \phi {x \circ y^{-1} }\) | \(=\) | \(\ds \map \phi x * \paren {\map \phi y}^{-1}\) | Definition of Identity Element |
- Result $(2)
- $:
\(\ds \map \phi y * \map \phi {y^{-1} \circ x}\) | \(=\) | \(\ds \map \phi {y \circ \paren {y^{-1} \circ x} }\) | Definition of Group Homomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {\paren {y \circ y^{-1} } \circ x }\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {e_G \circ x }\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x\) | Definition of Identity Element | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\map \phi y}^{-1} * \paren {\map \phi y * \map \phi {y^{-1} \circ x} }\) | \(=\) | \(\ds \paren {\map \phi y}^{-1} * \map \phi x\) | Group Axiom $\text G 3$: Existence of Inverse Element | ||||||||||
\(\ds \paren {\paren {\map \phi y}^{-1} * \map \phi y } * \map \phi {y^{-1} \circ x}\) | \(=\) | \(\ds \paren {\map \phi y}^{-1} * \map \phi x\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds e_H * \map \phi {y^{-1} \circ x}\) | \(=\) | \(\ds \paren {\map \phi y}^{-1} * \map \phi x\) | Definition of Inverse Element | |||||||||||
\(\ds \map \phi {y^{-1} \circ x}\) | \(=\) | \(\ds \paren {\map \phi y}^{-1} * \map \phi x\) | Definition of Identity Element |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.2$. Some lemmas on homomorphisms: Lemma $\text{(i)}$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Proposition $8.6$