# Group Homomorphism of Product with Inverse

## Theorem

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

Then:

$(1): \quad \forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
$(2): \quad \forall x, y \in G: \map \phi {y^{-1} \circ x} = \paren {\map \phi y}^{-1} * \map \phi x$

## Proof

Result $(1)$:

 $\displaystyle \map \phi {x \circ y^{-1} } * \map \phi y$ $=$ $\displaystyle \map \phi {\paren {x \circ y^{-1} } \circ y}$ Definition of Group Homomorphism $\displaystyle$ $=$ $\displaystyle \map \phi x$ $\displaystyle \leadsto \ \$ $\displaystyle \map \phi {x \circ y^{-1} } * e_H$ $=$ $\displaystyle \map \phi x * \paren {\map \phi y}^{-1}$ multiplying on the right by $\paren {\map \phi y}^{-1}$

Result $(2)$:

 $\displaystyle \map \phi y * \map \phi {y^{-1} \circ x}$ $=$ $\displaystyle \map \phi {y \circ \paren {y^{-1} \circ x} }$ Definition of Group Homomorphism $\displaystyle$ $=$ $\displaystyle \map \phi x$ $\displaystyle \leadsto \ \$ $\displaystyle e_H * \map \phi {y^{-1} \circ x}$ $=$ $\displaystyle \paren {\map \phi y}^{-1} * \map \phi x$ multiplying on the left by $\paren {\map \phi y}^{-1}$

$\blacksquare$