Hyperbolic Modules of Finite Group Algebras over Finite Fields of Characteristic Two

Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. We introduce \emph{$F$-special subgroups} and \emph{$F$-special elements} of $G$. In the case where $F$ contains a $p$th primitive root of unity for each odd prime $p$ dividing the order of $G$ (e.g. it is the case once $F$ is a splitting field for all subgroups of $G$), the $F$-special elements of $G$ coincide with real elements of odd order. We prove that a symmetric $FG$-module $V$ is hyperbolic if and only if the restriction $V_D$ of $V$ to every $F$-special subgroup $D$ of $G$ is hyperbolic, and also, if and only if the characteristic polynomial on $V$ defined by every $F$-special element of $G$ is a square of a polynomial over $F$. Some immediate applications to characters, self-dual codes and Witt groups are given.