Normality in Group Rings

Let KG be the group ring of a group G over a commutative ring K with unity. The rings KG are described for which xxσ = xσx for all x = ∑ g∈G αgg ∈ KG, where x → xσ = ∑ g∈G αgf(g)σ(g) is an involution of KG; here f : G → U(K) is a homomorphism and σ is an antiautomorphism of order two of G. Let R be a ring with unity. We denote by U(R) the group of units of R. A (bijective) map : R → R is called an involution if for all a, b ∈ R we have (a + b) = a + b , (ab) = b · a and a 2 = a. Let KG be the group ring of a group G over a commutative ring K with unity, let σ be an antiautomorphism of order two of G, and let f : G → U(K) be a homomorphism from G onto U(K). For an element x = ∑ g∈G αgg ∈ KG, we define x = ∑ g∈G αgf(g)σ(g) ∈ KG. Clearly, x → x is an involution of KG if and only if gσ(g) ∈ Ker f = {h ∈ G | f(h) = 1} for all g ∈ G. The ring KG is said to be σ-normal if (1) xx = xx for each x ∈ KG. The properties of the classical involution x → x∗ (where ∗ : g → g−1 for g ∈ G) and the properties of normal group rings (i.e., xx∗ = x∗x for each x ∈ KG) have been used actively for the investigation of the group of units U(KG) of the group ring KG (see [1, 2]). Moreover, they also have important applications in topology (see [7, 8]). Our aim is to describe the structure of the σ-normal group ring KG for an arbitrary order 2 antiautomorphism σ of the group G. Note that descriptions of the classical normal group rings and the twisted group rings were obtained in [1, 3] and [4, 5], respectively. The notation used throughout the paper is essentially standard. Cn denotes the cyclic group of order n; ζ(G) and CG(H) are the center of the group G and the centralizer of H in G, respectively; (g, h) = g−1h−1gh = g−1gh (g, h ∈ G); γi(G) is the ith term of the lower central series of G, i.e., γ1(G) = G and γi+1(G) = ( γi(G), G ) for i ≥ 1; Φ(G) denotes the Frattini subgroup of G. We say that G = A Y B is a central product of its subgroups A and B if A and B commute elementwise and, taken together, they generate G, provided that A ∩B is a subgroup of ζ(G). A non-Abelian 2-generated nilpotent group G = 〈a,b〉 with an antiautomorphism σ of order 2 is called a σ-group if G′ has order 2, σ(a) = a(a, b), and σ(b) = b(a, b). Our main result reads as follows. Theorem. Let KG be the noncommutative group ring of a group G over a commutative ring K and f : G → U(K) a homomorphism. Assume that σ is an antiautomorphism of order two of G such that x → x is an involution of KG. Put R(G) = {g ∈ G | σ(g) = g}. 2000 Mathematics Subject Classification. Primary 16S34.