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{\bf Generalized characters whose values on non-identity elements are roots of unity}\\
\emph{Geoffrey R. Robinson,\\
Institute of Mathematics,\\
University of Aberdeen,\\
Aberdeen AB24 3UE}
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\noindent {\bf Introduction:} Let $H$ be a finite group. We are concerned here with the question of when the
values taken by a generalized character $\theta$ of $H$ on non-identity elements are all roots of unity.
By elementary Galois theory and algebraic number theory, this is equivalent to the requirement that 
$|\theta(h)|^{2} = 1$ for all $h \in H^{\#}.$
Note that the generalized characters to be considered form an Abelian group under pointwise 
multiplication, which we done by $U(H).$
We will completely classify all such generalized characters
$\theta$ as long as the Sylow $2$-subgroup of $H$ does not have a cyclic subgroup 
of index $2$ (in particular, this allows the possibility that $H$
has odd order). It will turn out that (even if a Sylow $2$-subgroup of $H$ 
does have a cyclic subgroup of index $2)$, then
$\theta$ always factors as the product of a linear character $\lambda$ of $H$ and a generalized
character $\psi$ such that $\psi(h) \in \{1,-1\}$ for all $h \in H^{\#}.$  
When $H = O^{2}(H),$ this implies that $U(H)$ is the direct product
of $H/H^{\prime}$ with an elementary Abelian $2$-group. The main 
theorem of this note shows that (when $H = O^{2}(H)$ and a Sylow
$2$-subgroup of $H$ does not have a cyclic subgroup of index $2),$ 
the rank of the elementary Abelian $2$-group can be calculated directly from
the character table of $H.$