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Suppose we have a discrete group $G$, finite or infinite, on which we form the group algebra $\mathbb{F}_2[G]$. Suppose also that we have a map $S$:

$S : \mathbb{F}_2[G] \to \mathbb{F}_2[G]: \sum{a_g \cdot g} \to \sum{a_g \cdot g^{-1}}$

The problem I'm trying to solve is the following : given an element $x \in \mathbb{F}_2[G]$, find all $y \in \mathbb{F}_2[G]$ such that $y \cdot S(y) = x$.

For small groups $G$, we can write down a system of equations on the $a_g$ but I cannot find a general way... any help or lead appreciated !

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    The map induced by $g \mapsto g^{-1}$ is called the _classical involution_ in group ring theory. An element of a group ring is called _symmetric_ if it is fixed under the classical involution, and a unit of a group ring is called _unitary_ if the classical involution inverts it. Related questions were studied intensively by Victor Bovdi and others, you may find further references in MathSciNet and a collection of preprints in [arXiv](http://arxiv.org/). Hope that may give further leads.2013-07-01

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