Group rings with metabelian unit groups in characteristic 2

Abstract

Abstract. Let F be a field of characteristic p > 0 and let G be a non-abelian group. Denote by U(F G) the unit group of the group ring F G. In 1991, Shalev [6] provided a necessary and sufficient condition for U(F G) to be metabelian (i.e., for the commutator subgroup of U(F G) to be abelian) in the case where G is finite and p > 2. A few years later, Kurdics [4], as well as Coleman and Sandling [2] independently, extended this result to the case p = 2. Namely, they proved that when p = 2 and G is finite, U(F G) is metabelian if and only if either the commutator subgroup G′ of G is a central elementary abelian 2-group of order at most 4, or F is the field of 2 elements and G belongs to a specific class of non-nilpotent groups. For odd characteristic, the finiteness assumption on the order of G has already been relaxed: Catino and Spinelli [1] showed that the theorem of Shalev remains valid when G is a torsion group, and in 2022, Juhász and Spinelli [3] investigated the case where G contains elements of infinite order, discovering some new cases. In the case p = 2, Catino–Spinelli [1] and Mozgovoj [5] proved that the Kurdics, Coleman–Sandling theorem also holds for torsion groups. Furthermore, Mozgovoj’s work shows that the result remains valid even when G contains elements of infinite order, provided that G is non-nilpotent.