SOLOMON’S ZETA FUNCTION OF Bp(Cp3 )

Abstract

Abstract. In 1977, Solomon L. introduced a zeta function for orders for which all ideals of finite index must be known. This work follows our previous research [Zeta functions of Burnside rings of groups of order p and p 2 , Comm. Algebra, 37(2009Algebra, 37( ), 1758Algebra, 37( -1786, where we found the zeta function of the Burnside Ring for cyclic groups of prime order p and p 2 , respectively. The main objective of this paper is to obtain all ideals of finite index in Bp(C p 3 ) in order to determine ζ Bp(C p 3 ) (s) the zeta function of the Burnside Ring for a cyclic group of order p 3 . Mathematics Subject Classification (2010): 13C10, 11S40Keywords: Burnside rings, zeta functions, fiber product IntroductionThroughout this paper, G is a finite group. Its Burnside ring B(G) is the Grothendieck ring of the category of finite left G-sets. This is the free abelian group on the isomorphism classes of transitive left G-sets of the form G H for subgroups H of G, two such subsets being identified if their stabilizers H are conjugate in G; addition and multiplication are given by the disjoint union and Cartesian product, respectively.In Section 2, we recall the Burnside ring B (G) of a finite group G, along with the zeta function ζ B(G) (s) of B(G) and the ideals of a fiber product of rings.In Section 3, we recall the ideals of finite index in B p C p 2 according to [5], in order to compute the ideals of finite index in B p C p 3 via the fiber product of rings.Finally, in Section 4, we determine the zeta function ζ Bp(C p 3 ) (s) of the Burnside ring for a cyclic group C p 3 . Preliminaries2.1. Burnside rings. Let X be a finite G-set and let [X] be its G isomorphism class. We define [H] belongs to the set of conjugacy classes of subgroups of G, which we call C (G) . That isFor further information about the Burnside ring, see [1].Let H ≤ G be a subgroup and X a G-set, we denote the set of fixed points of X under the action of H byWe define the mark of H on X as the number of elements of X H and we call itWe define B (G) := [H]∈C(G) Z, thus we have the following mapwhich is a morphism of semirings that extends to a unique injective morphism of Solomon's zeta function. Let R be a Dedekind domain with quotient fieldK, and let B be a finite dimensional K − algebra. For any finite dimensionalAn R − order in B is a subring Λ of B such that the center of Λ contains R and such that Λ is a full R − lattice in B.Let p ∈ Z be a rational prime and let Z p be the ring of p − adic integers. We denote the following tensor products bywhere we have that B p (G) is a Z p − order, being B p (G) its maximal order. For further information about orders, see [3, Chapters 2 and 3].Definition 2.2. We define the Solomon's zeta function ζ Λ (s) of an order Λ, as follows:which is a generalization of the classical Dedekind zeta function ζ K (s) of an algebraic number field K.For the commutative rings B p (G) and B p (G), the sum extends over all the ideals of finite index and converges uniformly on compact subsets of {s ∈ C : Re(s) > 1} . For fu...