Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2

Theorem

Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.

Let $\circ: G \times G \to G$ be the binary operation defined as:

$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

$\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$.

To prove $G$ is isomorphic to $\struct {\R, +}$, it is sufficient to find a bijective homorphism $\phi: \to \R \to G$:

$\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$

the identity element of $G$ is $0$

the inverse of $x$ in $G$ is $-x$.

This also holds for $\struct {\R, +}$.

This hints at the structure of a possible such $\phi$:

$(1): \quad$ that it is an odd function

$(2): \quad$ that it passes through $0$

$(3): \quad$ that its image is the open interval $\openint {-1} 1$.

One such function is the hyperbolic tangent function $\tanh$, which indeed has the above properties on $\R$.

Then we have that:

$\ds \map {\tanh } {x + y}$

$=$

$\ds \dfrac {\tanh x + \tanh y} {1 + \tanh x \tanh y}$

$\ds $

$=$

$\ds \tanh x \circ \tanh y$

This demonstrates the homorphism between $\struct {\R, +}$ and $\struct {G, \circ}$.

Hence the result that $\struct {\R, +} \cong \struct {G, \circ}$.

$\blacksquare$