SYMMETRIC SUBGROUPS OF THE UNIT GROUP
OF THE MODULAR GROUP ALGEBRA OF A FINITE p-GROUP
Tsapok A.G.
Zaporozhye State University, Zaporozhye
Let V(KG) be the normalized unit group of the modular group algebra of a finite
p-group G over the field K of p elements. We introduce a notion of symmetric
subgroup in V(KG) as a subgroup which is invariant under the action of the classical
involution of the group algebra KG. We study properties of symmetric subgroups and
construct a counterexample to the conjecture by V. Bovdi, which states that
V ( KG)  G, S* ,
(1)
where S* is a set of symmetric units of V(KG).
Let H be a subset of the group algebra KG. We call H symmetric if H*=H, where
H *  h* | h  H . If H is a subgroup of V(KG) than we call it symmetric subgroup if
H*=H. Some properties of symmetric subgroups were formulated and proved in our
research.
Note that the subgroup G, S* from the conjecture (1) is symmetric. Using the
computer algebra system GAP [1] and the package LAGUNA [2], we discovered that
the quaternion group of order 8 gives a counterexample to this conjecture. We also
give purely theoretical proof of this result, calculating that the order of the subgroup
G, S* is 64, while V KG   128 .
References:
1. The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.3,
Aachen, St Andrews, 2002 (http://www.gap-system.org).
2. Bovdi V., Konovalov A., Schneider C. and Rossmanith R.: GAP 4 package
LAGUNA — Lie AlGebras and
UNits of group Algebras, Version 3.1, 2003
(http://ukrgap.exponenta.ru/LAGUNA.htm).