Can an element other than the neutral element be its own inverse? Take the following operation $*$ on the set $\{a, b\}$:


*

*$a * b = a$

*$b * a = a$

*$a * a = b$

*$b * b = b$


$b$ is the neutral element. Can $a$ also be its own inverse, even though it's not the neutral element? Or does the inverse property require that only the neutral element may be its own inverse but all other elements must have another element be the inverse.

Denoting 1 for the identity element, we have for every group element $a$:
$a=a^{-1}\Leftrightarrow a^2=1$.
So a non-identity element is its own inverse iff it has order 2. This is perfectly possible, as Gerry Myerson showed. Or look at the group of automorphisms of $\mathbb{C}-\{0\}$ (invertible ring homomorphisms), the identity element being the identity map, and group multiplication being composition of automorpisms. Then complex conjugation has order 2: it sends $x+iy$ to $x-iy$, and the latter is sent back to $x+iy$.
