We prove that the double covers of the alternating and symmetric groups
are determined by their complex group algebras. To be more precise, let
be an integer,
a finite group,
and let
and
denote the double
covers of
and
, respectively.
We prove that
if and only if
,
and
if
and only if
or
.
This in particular completes the proof of a conjecture proposed by the second and
fourth authors that every finite quasisimple group is determined uniquely up to
isomorphism by the structure of its complex group algebra. The known results on
prime power degrees and relatively small degrees of irreducible (linear and
projective) representations of the symmetric and alternating groups together
with the classification of finite simple groups play an essential role in the
proofs.
Keywords
symmetric groups, alternating groups, complex group
algebras, Schur covers, double covers, irreducible
representations, character degrees
Institut für Algebra, Zahlentheorie
und Diskrete Mathematik
Fakultät für Mathematik und Physik
Leibniz Universität Hannover
Welfengarten 1
D-30167 Hannover
Germany