Zeta Function of the Burnside Ring for Cyclic Groups

International Journal of Algebra, Vol. 5, 2011, no. 26, 1255-1266 Zeta Function of the Burnside Ring for Cyclic Groups David Villa-Hernandez Instituto de Matematicas, UNAM (Campus Morelia) Morelia, Michoacan, Mexico dvilla@matmor.unam.mx Abstract. The purpose of this article is to determine the zeta function ζ B(G) (s) of the Burnside ring B(G), for cyclic groups. Mathematics Subject Classification: 20C05, 11M06 Keywords: Burnside rings; Zeta functions; Fiber product 1. INTRODUCTION. According to the definition given by Solomon for the zeta function of an order, it is necessary to know all its ideals of finite index, which might be complicated. In this work we present a method used by Bushnell-Reiner [6, section 2], which depends only on the finite set of the isomorphism classes of the ideals of finite index. We will use this method for the zeta function of Burnside Ring for cyclic groups. We can see that for n =1, 2 this is a better alternative than the method used in [8, sections 2,3], where the same results are obtained by computing all the ideals and doing the direct calculation of the zeta functions. We will consider a finite group G. Let X be a finite G-set and let X be its G isomorphism class. We define B + (G) := { X X a finite G-set }, which is a commutative semiring with unit, with the binary operations of disjoint union and Cartesian product. Definition 1.1. We define the Burnside ring B(G)of G as the Grothendieck ring of B + (G). We observe that as an abelian group, B(G) is free, generated by elements of the form G H, where H belongs to the set of conjugacy classes

1256 D. Villa-Hernandez of subgroups of G, which we call C(G). That is B(G) = Z(G H). H C(G) For 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 by X H = {x X h x = x, h H}. We define the mark of H on X as the number of elements of X H and we call it ϕ H (X). In [3] the reader can find some of the properties satisfied by ϕ H. Let X be a G-set. We define B(G) := H C(G) Z; thus we have the following map ϕ : B + (G) B(G) X (ϕ H (X)) H C(G), which is a morphism of semirings that extends to a unique injective morphism of rings ϕ : B(G) B(G). Let q Z be a prime integer. We have the following set: { a } Z {q} = b Q : q b, so we denote the following tensor products by B {q} (G) =Z {q} B(G) and B{q} (G) =Z {q} Z Z B(G) where we have that B {q} (G) isanz {q} -order, and for which B {q} (G) is its maximal order. For further information about orders, see [3, 5]. Definition 1.2. We define the zeta function ζ Λ (s) of an order Λ, as follows: ζ Λ (s) := Λ I s, I Λ, ideal (Λ : I) < which, in case B(G), B(G),B {q} (G) and B {q} (G), converges uniformly on compact subsets of {s C Re(s) > 1}. See [7]. Notation 1.3. Let B p be a Q {p} -algebra in the field of p-adic numbers, Γ p a Z {p} -order, V p a B p -module and let M p,n p be two full Γ p -lattices in V p. We

define Zeta function of Burnside ring 1257 Z Γp (M p ; s) = (Γ p : N p ) s. N p Γ p N p = Mp Therefore, ζ Γp (s) = M p Z Γp (M p ; s), where this finite sum extends over all the representatives of the isomorphism classes of the full Γ p -lattices in V p. Let A p = End Bp V p and Λ p = End Γp M p, whence A p = Aut Bp V p and Λ p = Aut Γp M p. We define the conductor of M p in N p as {M p,n p } = {x A p : M p x N p }, which turns out to be a full Z {p} -lattice in A p. Let X, Y be two full Z {p} -lattices in V p. We define (X : Y )= (X : X Y ) (Y : X Y ). Let x A p and N p be any full Z {p} -lattice in V p. We define the following norm x Vp =(N p x : N p ) which is multiplicative and does not depend on N p. Furthermore, we can see that x Vp = 1 whenever x belongs to the units of a Z {p} -order in A p. We observe that A p inherits a topology form its structure as a finite dimensional Q {p} -space. We choose a Haar measure d x on A p. Then we have: Z Γp (M p ; s) =μ ( ) Λ 1 p (Γ p : M p ) s Φ {Mp,Γp} (x) x s V p d x. For further details on this result, see [2, 2.1]. For further information about the zeta function, see [2, 4] We assume that f 2 A p A A 2 f 1 g 2 A 1 A g 1

1258 D. Villa-Hernandez is a fiber product of rings, where all the maps are ring surjections. By definition A = {(a 1,a 2 ): a i A i for i =1, 2 and g 1 (a 1 )=g 2 (a 2 )}. Let I A and I i A i be left ideals, such that I i = f i (I) for i =1, 2. Let A 2 be a PID. Then I 2 = A 2 β. We have: α I 1 such that (α, β) I. Let J = {c A 1 :(c, 0) I}, which is an ideal of A 1. We have that I = A (α, β)+(j, 0), that satisfies: 1. f 2 (I) =A 2 β, 2. g 1 (J) =0, and 3. g 1 (α) =g 2 (β). For further details on this result, see [6]. 2. IDEALS OF FINITE INDEX IN B {p} (C p n). Let n N and Γ n = B {p} (C p n) be the Burnside ring of the cyclic group of order p n. We have that the conjugacy classes of C p n are { Cp n,pc p n,p 2 C p n,..., p n C p n =1 }, whence a basis for Γ n is {a 0 = C p n/c p n,a 1 = C p n/pc p n,...,a n = C p n/p n C p n = C p n/1}. Therefore, Γ n = Z {p} Z{p} a 1... Z{p} a n. Furthermore Γ n = Z (n+1) {p} is its maximal order On the other hand, we known that { G/K for H K ϕ H (G/K) = 0 for H K, and then, for C(G) ={C p n,pc p n,p 2 C p n,..., p n C p n}, we have that ϕ induces the following inclusion ϕ Γ n Γ n X (ϕ H (X)) H C(G) a 0 (1,..., 1) a 1 (0,p,...,p) a 2 (0, 0,p 2,...,p 2 )..... a n (0,...0,p n )

Zeta function of Burnside ring 1259 Therefore, we can see Γ n in Γ n as follows: { } Γ n = (u 0,...,u n ) Z (n+1) {p} : (u l u l 1 ) p l Z {p} for l =1,...,n Γ n, to which we can give the following fiber product structure: (u 0,...,u n ) u n f 2 Γ n Z {p} f 1 g 2 Γ n 1 Z {p} /p n Z {p} g 1 (u 0,...,u n 1 ) u n 1 = u n We observe that Z {p} is a PID. Therefore, it has ideals of the form p t Z {p}, for every integer t 0, and according to the structure of the fiber product, we have that the ideals of finite index in Γ n are ideals of the form I = ( α, p t) Γ n +(J, 0) where α is an element of any ideal of Γ n 1 and J Γ n 1 is an ideal such that: 1.g 1 (J) = 0, and 2.g 1 (α) =g 1 (p t ). 3. THE ZETA FUNCTION FORζ B{p} (C p )(s). From the last section, for n =1, we have that the ideals of finite index in Γ 1 = B {p} (C p ) are ideals of the form where: 1.g 1 ( p m Z {p} ) =0, and 2.g 1 ( p k ) = g 1 ( p l ). Therefore, I = ( p k,p l) Γ 1 + ( p m Z {p}, 0 ), I 1 = p m Z {p} p l Z {p} for 1 m, l I = { (p I 2 = k u, p l v ) } Z 2 {p} :(v u) pz {p} for k = l =0 or 1 k, l We observe that the only isomorphism classes of the fractional ideals of finite index of Γ 1 are Γ 1 and Γ 1. Hence ζ Γ1 (s) =Z Γ1 (Γ 1 ; s)+z Γ1 (Γ 1; s).

1260 D. Villa-Hernandez ( 2 Observation 3.1. We choose a Haar measure d x on Q{p}) such that d x =(d α) 2, where d α is a Haar measure on Q {p}, such that d α =1. Z p Therefore, we have: μ (Γ 1 )=1. Furthermore, Γ 1 is local, where radγ 1 = ( ) 2 pz{p}. ( ) Thus Γ 1 =Γ 2 1 pz{p}. (1). For Γ 1, we have: {Γ 1, Γ 1 } =Γ 1 and Aut Γ1 Γ 1 =Γ 1, whence μ (Aut Γ1 Γ 1 ) 1 = (Γ 1 :Γ 1 )=p 1. Thus Z Γ 1 (Γ 1 ; s) = (p 1) Q {p} 2 T h Γ 1 x s Q 2 d x = ( 1 2p s + p 1 2s) ζ Γ {p} 1 (s), U (pz{p}) 2i where ζ Γ 1 (s) =1/ (1 p s ) 2. (2). For Γ 1, we have: {Γ 1, Γ 1} = ( ) 2 pz {p} and AutΓ1 Γ 1 =Γ 1, whence Furthermore, Hence μ (Aut Γ1 Γ 1 ) 1 =1. (Γ 1 :Γ 1)= NCp (C p ) /C p NCp (pc p ) /pc p = p. Z Γ1 (Γ 1; s) =p s x s Q 2 {p} 2 T Q {p} (pz{p}) 2 Finally, from the previous points, we obtain: ζ Γ1 (s) = ( 1 p s + p 1 2s) ζ Γ 1 (s). d x = p s ζ Γ 1 (s). This result is obtained by using another method in [8, section 2] 4. THE ZETA FUNCTION FOR ζ B{p}(C p 2) (s). From the section 2, for n =2, we have that the ideals of finite index in Γ 2 = B {p} (C p 2) are ideals of the form I = ( α, p t) Γ 2 +(J, 0) where α is an element of any ideal of Γ 1 and J Γ 1 is an ideal such that 1.g 1 (J) =0 2.g 1 (α) =g 1 (p t ).

Zeta function of Burnside ring 1261 Therefore I = {( p m u 0 x + p k u, p n (x + py)+p l v, p t (x + py + p 2 z) ) Z 3 {p} : x, y, z Z {p}, } { i) a) u, v Z {p}, 1 k, 2 l b) u, v Z {p} such that (v u) pz {p}, 1 k, 2 l and ii) a) (u 0 1) pz {p},n= m = t =0 b) u 0 Z {p},n= t =1, 1 m c) (u 0 1) pz {p}, n=t=1, 1 m d) u 0 Z {p}, 2 n, t, 1 m e) (u 0 1) pz {p}, 2 n, t, 1 m whence the only isomorphism classes of fractional ideals of Γ 2 are M 1 =Γ { 2, M 2 =Γ 2, M 3 = Z {p} Γ1, M } 4 =Γ 1 Z{p}, M 5 = (x, y, z) Z 3 {p} : (z x) pz {p}, } M 6 = {(x, y, z) Z 3{p} : (y x) pz {p}, (z y) pz {p}, } M 7 = {(x, y, z) Z 3{p} : (z y) p2 Z {p}, } M 8 = {(x, y, z) Z 3{p} : px y + z p2 Z {p}, { } M 9 = (x, y, z) Z 3 {p} : x y + z pz {p}. Thus ζ Γ2 (s) = 9 Z Γ2 (M i ; s). i=1 ( 3 Observation 4.1. We choose a Haar measure d x on Q{p}), such that d x = (d α) 3, where d α is a Haar measure of Q {p} such that d α = Z p 1. Hence we consider μ (Γ 2 )=1. Furthermore, Γ 2 is local, where radγ 2 = ( ) pz {p} pγ1. Thus Γ 2 =Γ 2 pz{p} pγ1. (1). For Γ 2, we have: {Γ 2, Γ 2 } =Γ 2 and Aut Γ2 Γ 2 =Γ 2, whence μ (Aut Γ2 Γ 2 ) 1 =(Γ 2 :Γ 2)=p(p 1) 2. Thus Z Γ2 (Γ 2 ; s) =p(p 1) 2 d x = [Γ 2 U (pz{p} L pγ1)]\{0} [1 3p s +3p 2s +( 1 p + p 2 ) p 3s +2(1 p) p 1 4s +(p 1)p 2 5s ] (1 p s ) 3, where ζ Γ 2 (s) =1/

1262 D. Villa-Hernandez (2). For Γ 2, we have: {Γ 2, Γ 2} =(p, p 2,p 2 )Γ 2 and Aut Γ 2 Γ 2 =Γ 2, whence μ (Aut Γ2 Γ 2 ) 1 =1. Furthermore, (Γ 2 :Γ 2)=p 3. Hence Z Γ2 (Γ 2 ; s) =p3s L p 2 Z {p} L p 2 Z {p})\{0} d x = p 2s (3). For M 3, we have: {M 3, Γ 2 } =(p, p,, p)m 3 and Aut Γ2 M 3 = M3, whence μ (Aut Γ2 M 3 ) 1 = (Γ 2 : M3 ) = p 1. Furthermore, (M 3 :Γ 2 ) = p 2. Thus Z Γ2 (M 3 ; s) = (p 1)p 2s L pγ1)\{0} d x = p s (1 2p s + p 1 2s ) (4). For M 4, we have: {M 4, Γ 2 } =(p, p 2,p 2 )Γ 2 and Aut Γ2 M 4 = M4, whence μ (Aut Γ2 M 4 ) 1 = (Γ 2 : M4 ) = p 1. Furthermore, (M 4 :Γ 2 ) = p 2. Thus Z Γ2 (M 4 ; s) = (p 1)p 2s L p 2 Z {p} L p 2 Z {p})\{0} x s d x = Q 3 {p} (p 1)p 3s (5). For M 5, we have: {M 5, Γ 2 } =(p, p 2,p 2 )Γ 2 and Aut Γ 2 M 5 = M5, whence μ (Aut Γ2 M 5 ) 1 = (Γ 2 : M5 ) = p 1. Furthermore, (M 5 :Γ 2 ) = p 2. Thus Z Γ2 (M 5 ; s) = (p 1)p 2s L p 2 Z {p} L p 2 Z {p})\{0} x s d x = Q 3 {p} (p 1)p 3s (6). For M 6, we have: {M 6, Γ 2 } =(p, p,, p)m 3 and Aut Γ2 M 6 = M 6, whence μ (Aut Γ2 M 6 ) 1 =(Γ 2 : M 6 )=(p 1) 2. Furthermore, (M 6 :Γ 2 )=p. Thus Z Γ2 (M 6 ; s) =(p 1) 2 p s L pγ1)\{0} d x = (p 1)p 2s (1 2p s + p 1 2s ) (7). For M 7, we have: {M 7, Γ 2 } =(p, p,, p)m 3 and Aut Γ2 M 7 = M7, whence μ (Aut Γ2 M 7 ) 1 =(Γ 2 : M7 )=p(p 1). Furthermore, (M 7 :Γ 2 )=p. Thus Z Γ2 (M 7 ; s) =

Zeta function of Burnside ring 1263 p(p 1)p s L pγ1)\{0} d x = p1 2s (1 2p s + p 1 2s ) (8). For M 8, we have: {M 8, Γ 2 } =(p, p,, p)m 3 and Aut Γ2 M 8 =Γ 2, whence μ (Aut Γ2 M 8 ) 1 =(Γ 2 :Γ 2)=p (p 1) 2. Furthermore, (M 8 :Γ 2 )=p. Thus Z Γ2 (M 8 ; s) =p(p 1) 2 p s d x = L pγ1)\{0} p(p 1)p 2s (1 2p s + p 1 2s ) (9). For M 9, we have: {M 9, Γ 2 } =(p, p 2,p 2 )Γ 2 and Aut Γ 2 M 9 = M6, whence μ (Aut Γ2 M 9 ) 1 =(Γ 2 : M6 )=(p 1) 2. Furthermore, (M 9 :Γ 2 )=p 2. Thus Z Γ2 (M 9 ; s) = (p 1) 2 p 2s L p 2 Z {p} L p 2 Z {p})\{0} d x = (p 1)2 p 3s From the nine points above, we finally obtain ζ Γ2 (s) = (1 2p s +(1+p + p 2 )p 2s 2p 1 3s +(1 p + p 2 )p 1 4s +(p 1)p 2 5s ) This result is obtained by using another method in [8, section 3]. Observation 4.2. We observe that the method used in the last two sections, provides us with a recursive method to compute the Zeta Function of B {p} (C p n), for 0 n N. 5. THE ZETA FUNCTION FOR ζ B{p} (P Q)(s). Lemma 5.1. Let G = P Q be a group, which is the direct product of P and Q, where P G is the Sylow p-subgroup of G and Q G is a subgroup of order prime relative to p. We have the following ring isomorphism: B {p} (G) = B {p} (P ), H C(Q) where this product extends over all conjugacy classes of subgroups of Q.

1264 D. Villa-Hernandez Proof. We know that B {p} (G) = K C (G) O {p} (K) =K B {p} (G) e {p} G,K, where e {p} G,K are the primitive idempotents of B {p} (G). Furthermore, for each subgroup K G, we have: K = T H, where T P and H Q. Since O {p} (K) is the minimum normal subgroup of K, such that K/O {p} (K) isa p-group, the condition O {p} (K) =K implies that K = {1} H. We denote this subgroup by H, hence we have B {p} (G) = H C(Q) B {p} (G) e {p} G,H. Therefore, we only have to prove that there is a ring isomorphism between B {p} (G) e {p} G,H and B {p} (P ) for each H C (Q). Remember that B {p} (P )= L C(P ) Z {p} (P/L). Furthermore, from [9, Theorem 3.1] we have: B {p} (G) e {p} G,H = Z {p} (G/M) e {p} G,H. M C (G) O {p} (M) =H Since the condition O {p} (M) =H implies that M = T H, for T C (P ), then B {p} (G) e {p} G,H = Z {p} (G/ (T H)) e {p} G,H. T C(P ) In order to conclude this proof, we only need to observe that the following is a ring isomorphism: T C(P ) Z {p} (G/ (T H)) e {p} G,H α = L C(P ) Z {p} (P/L) (G/ (T H)) e {p} G,H w Q (H)(P/T), for w Q (H) = N Q (H) /H. It is clear that α is a bijection of abelian groups, also that α(1) = 1 and α(ab) = α(a)α(b). This concludes the proof. Corollary 5.2. Let G = P Q be a group, which is the direct product of P and Q, where P G is the Sylow p-subgroup of G and Q G is a subgroup

Zeta function of Burnside ring 1265 of order prime relative to p. Therefore, the following relation is fulfilled: ζ B{p} (G) (s) = ζ B{p} (P ). (s) H C(Q) The proof of this corollary is clear from the previous lemma and from the properties of zeta function. Observation 5.3. As a particular case of the previous corollary, let C be a cyclic group of order p n m, where n, m N and p Z is a prime such that p m. Let m = q α 1 1 qαr r be the prime factorization of m. If C p n is the cyclic group of order p n, denoting by Γ = B {p} (C) the Burnside ring of C, and by Γ n = B {p} (C p n) the Burnside ring of C p n. Hence we obtain the following ring isomorphisms: 1. Γ = Γ Q r i=1 (α i+1) n 2. Γ (n+1) Q r = Z i=1 (α i+1) {p}, whence ζ Γ (s) =[ζ Γn (s)] Q r i=1 (αi+1). Acknowledgment. This article is part of my PhD thesis under the supervision of Prof. Alberto Gerardo Raggi Cardenas. I am grateful to him for many valuable suggestions. References [1] Bouc, S. Burnside rings. (2000). Handbook of algebra, vol 2, 739-804, North- Holland, Amsterdam. [2] Bushnell, C. J., Reiner, I. (1987). Zeta functions of arithmetic orders and Solomon s conjectures. Math. Z. 173, 135-161. [3] Curtis, C. W., Reiner, I. (1987). Methods of representation theory with applications to finite groups and orders. Vol. 2, Chapter 2. Wiley-Interscience, New York, N. Y. [4] Raggi, C. A. G. (1986). Zeta functions of tow-sided ideals in arithmetic orders. Math. Z. 192, 353-382. [5] Reiner, I.(1975). Maximal Orders, London-New York. Academic Press. [6] Reiner, I. (1980). Zeta functions of integral representations. Communication in Algebra 8, pp.911-925 [7] Solomon, L. (1977). Zeta Functions and Integral Representation Theory. Advances in Mathematics 26, 306-326. [8] Villa-Hernandez D. (2009). Zeta functions of Burnside rings of grups of order p and p 2. Communications in Algebra, 37. pp 1758-1786.

1266 D. Villa-Hernandez [9] Wolfgang Kimmerle, Florian Luca and Gerardo Raggi (2008). Irreducible components and isomorphisms of the Burnside ring. Journal of Group Theory, Vol. 11, Number 6. pp 831-844. Received: May, 2011