Order of a group element ($\mathrm{ord}(g)$)

The order of a group element $g$ is the least positive integer $n$ for which $g^n$ is the identity. If no such integer exists, $g$ has infinite order.

A group element has infinite order when no positive power of it is the identity element.

An involution is a group element $s$ satisfying $s^2=1$. Some conventions require $s\ne 1$.

Let $F$ be a linear map on a vector space over a field of characteristic different from two, with $F^2=I$. ThenThe projections satisfy $P_{+}{P}_{-}=0$ and $P_{+}+P_{-}=I$, giving existence and uniqueness of the decomposition. In characteristic two, the two signs coincide and the conclusion need not hold.