Group/Examples/x+y over 1+xy

Theorem
The set of real numbers $$\left\{{x \in \reals: -1 < x < 1}\right\}$$, under the operation:
 * $$x \circ y = \frac {x + y} {1 + x y}$$

is a group.

Proof
Let $$-1 < x, y, z < 1$$.

We check the group axioms in turn:

G1: Associativity
$$ $$

$$ $$ $$

G2: Identity
$$ $$ $$ $$

Similarly, putting $$y = 0$$ we find $$x \circ y = x$$.

So $$0$$ is the identity.

G3: Inverses
$$ $$

Similarly, putting $$x = -y$$ gives us $$\left({-y}\right) \circ y = 0$$.

So each $$x$$ has an inverse $$-x$$.

G0: Closure
First note that: $$-1 < x, y < 1 \implies x y > -1 \implies 1 + x y > 0$$.

Next:

$$ $$ $$ $$ $$ $$ $$

Finally:

$$ $$ $$ $$ $$ $$ $$

Thus $$-1 < x, y < 1 \implies -1 < x \circ y < 1$$, and we see that in this range, $$\circ$$ is closed.

Thus the given set and operation form a group.