Symmetric Units in Modular Group Algebras

Let p be a prime, G a locally finite p -group, K a commutative ring of characteristic p . The anti-automorphism g 7→ g of G extends linearly to an anti-automorphism a 7→ a∗ of KG . An element a of KG is called symmetric if a∗ = a . In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group. Let G be a group, K a commutative ring (with 1), and U(KG) the group of units in the group algebra KG . The anti-automorphism g 7→ g of G extends linearly to an anti-automorphism a 7→ a∗ of KG ; this extension leaves U(KG) setwise invariant. An element a of KG is called symmetric if a∗ = a . 1991 Mathematics Subject Classification. Primary 16S34. V. Bovdi is indebted to the University of Alberta for warm hospitality and generous support during a period when part of this work was done. His research was also supported by the Hungarian National Foundation for Scientific Research grant No. T 014279.