Originally Posted by

**wsldam** By the way here are the proofs I roughed out (e is the identity and z is the inverse):

For identity:

$\displaystyle a \star e = a$ (known)

$\displaystyle (a \star e) \star a = a \star a$

$\displaystyle a \star (e \star a) = a \star a$ (associative)

Because a=a, this implies $\displaystyle e \star a = a$ (possible hand waving)

For inverse:

$\displaystyle a \star z = e$ (known)

$\displaystyle (a \star z) \star a = e \star a$

$\displaystyle (a \star z) \star a = a \star e$ (by previous hand waving)

$\displaystyle a \star (z \star a) = a \star e$ (associative)

Because a=a, this implies $\displaystyle z \star a = e$ (more possible hand waving)