Group Homomorphism of Product with Inverse

From ProofWiki
Jump to navigation Jump to search

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

Retrieved from "https://proofwiki.org/w/index.php?title=Group_Homomorphism_of_Product_with_Inverse&oldid=628279"