Polar decomposition of a group

Origin of the concept
The concept of polar decomposition can be see, for instance, in the polar decomposition of complex numbers, and the polar decomposition of linear operators on complex vector spaces.

Origin of the term
The notion of polar decomposition for a general group was introduced by Tuval Foguel, in his paper Polar Decomposition of Locally Finite Groups.

Definition
A polar decomposition of a group $$G$$ is the following datum:

An involutive automorphism $$\tau \in Aut(G)$$ (viz an automorphism $$\tau$$ satisfying $$\tau^2 = Id$$) such that the following holds.

satisfying the following compatibility condition:

Let $$P(\tau)$$ be the set of all elements of the form $$g\tau\left(g^{-1}\right)$$. Then every element of $$P(\tau)$$ has a unique squareroot in $$P(\tau)$$. (It may have other square-roots outside $$P(\tau)$$).

We say that $$G$$ has a polar decomposition due to $$\tau$$, or equivalently that $$(G,\tau)$$ is the datum of a group with a polar decomposition.

A polar decomposition makes $$G$$, naturally, into a product of the following two subsets:


 * The stabilizer of $$\tau$$, viz those elements $$g$$ such that $$\tau(g) = g$$
 * The set $$P(\tau)$$ of all elements that can be written as $$g\tau\left(g^{-1}\right)$$