Symmetric units in modular group algebras

Abstract

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g → g −1 of G extends linearly to an anti-automorphism a → 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 → g −1 of G extends linearly to an anti-automorphism a → a * of KG ; this extension leaves U (KG) setwise invariant. An element a of KG is called symmetric if a * = a.

On Symmetric Units in Group Algebras

“…Obviously, t , (g − α)(g −1 − α) and (h − β)(h −1 − β) are symmetric units, and since symmetric units commute, so t , α(g + g −1 ) and β(h + h −1 ) commute as well. Since αβ = 0 , we reached the requived conclusion in the second paragraph of [3] (p.805). Furthermore, for a Gfavourable K , the symmetric units of KG form a multiplicative group if and only if every pair of elements of…”

“…By Lemma 3 the elements of odd order in G form an abelian subgroup A , and either A is central or [G : C G (A)] = 2 and u Assume now that A is central in G. Then G = A × P and according to [3] the 2 -Sylow subgroup P is a direct product of an elementary 2 -group and one of those H which were mentioned in the conditions 3.(i)-3. (iv) of Theorem 1.…”

“…The problem when the set of symmetric units form a group was first studied for the case of rings in the paper of Lanski [7], and for group algebras in [3,4].…”

Symmetric Units in Alternative Loop Rings

“…is an abelian subgroup of index 2 and t −1 ct = c −1 for any c ∈ C and t / ∈ C. By [1], G = 16Γ 2 c 2 is admissible. In particular, elements of order 2 in G are central.…”

WHEN IS A UNIT LOOP f-UNITARY?

“…This situation has received quite a bit of attention recently. Bovdi and Kovacs determined when U * (KG) (K a field) is normal in U(KG) [4]. Bovdi and Erdei have considered the possibility that a group may have a normal complement in the unitary group U * (ZL) [3].…”