Number of self-inverses in a group Is there some established name for the number of self-inverses in a group (or if we generalize, for the cardinality of the set of self-inverses)?
For example, in $\mathbb R$ under addition, that number is $1$, since $0$ is the only self-inverse. In $\mathbb R$ under multiplication, that number is $2$, since the self-inverses are $-1$ and $1$. That number is $4$ for the group
* | e  u  v  w
--+------------
e | e  u  v  w
u | u  e  w  v
v | v  w  e  u
w | w  v  u  e

Note: it seems this number has some properties, like it can't be $3$. I welcome a reference on these properties.
 A: An element of order $2$ in a group is often called an involution. In geometric groups reflections are involutions, as are half turns and central involutions, for example.
Involutions are normally distinguished from the identity element, which is also "self-inverse".
To show that if a finite group has an involution, the number of self-inverse elements is even, first note that the number of elements of a group with an element of order $2$ is even. And then pair each element which is not self-inverse with its inverse - that takes out an even number of elements, leaving an even number to be self-inverse (and an odd number of involutions).
If a group has odd order it has just one self-inverse element - the identity.
Trust this helps. The theory around involutions was fundamental in the classification of finite simple groups. Of course that is abstract, but I mentioned the geometric ideas because these motivated a significant approach to the problem. What I have given above is basic and well-known. The more advanced theory is definitely more challenging. But geometric reflection groups might be an area to explore further. Look for Coxeter groups for further references.
