Zeta Functions of Burnside Rings for Symmetric and Alternating Groups

International Journal of Algebra, Vol. 6, 2012, no. 25, 1207-1220 Zeta Functions of Burnside Rings for Symmetric and Alternating Groups David Villa-Hernández Benemérita Universidad Autónoma de Puebla Facultad de Ciencias Fisico Matemáticas dvilla@fcfm.buap.mx Abstract. The purpose of this article is to study the zeta function of the Burnside Ring for some symmetric and alternating groups. Based on the soluble components of the Burnside Ring, we will clearly compute the zeta function of the Burnside Ring for the groups S 3,A 4 and A 5 in the local and global cases, and for the group S 4 in the local case p =3. Mathematics Subject Classification: 20C05, 11M06 Keywords: Burnside rings; zeta functions 1. Introduction. Throughout this paper, let G be a finite group. Let X be a finite G set and let [X] be its G isomorphism class. Definition 1.1. We define the Burnside ring B(G) ofg as the Grothendieck ring of 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, that we call [X] +[Y ]:=[X Y ] and [X] [Y ]:=[X Y ] respectively. Further, 0 := [ ] and 1 := [G G]. We observe that as an abelian group, the Burnside ring B(G) is free, generated by elements of the form [G H], where H belongs to the set of conjugacy

1208 D. Villa-Hernández classes of subgroups of G, which we call C(G). That is: B(G) = Z [G H]. H C(G) Let H, K be two subgroups of G. Let H R K be the set of representatives of induced partition in G by the double lateral classes of the form HgK, then, [G H] [G K] = [ G (H gkg 1 ) ]. g H R K 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). Some of the properties that satisfy ϕ H, are: i). ϕ H (X Y )=ϕ H (X)+ϕ H (Y ) for every X and Y two G sets. ii). ϕ H (X Y )=ϕ H (X)ϕ H (Y ) for every X and Y two G sets. iii). ϕ H (G H) = W (H) the order of the Weyl group of H, where W (H) = N G (H) /H, for N G (H) the normalizer of H in G. iv). Let H and K be subgroups of G in the same conjugacy class in C(G), then, ϕ H (X) =ϕ K (X), for every X a G set. v). ϕ K (G H) =0 K is not a subconjugate of H in C(G). vi). ϕ K (G H) = W (H) {E G : E = G H y K E}. For further information about marks, see [2]. Definition 1.2. Let X be a G set. We define B(G) := Z. H C(G)

We have that the application Zeta functions of Burnside rings 1209 ϕ : B + (G) B(G) [X] (ϕ H (X)) H C(G) is a morphism of semirings. This application can be extended specifically to an injective morphism of rings ϕ : B(G) B(G). Let p Z be a rational prime and let Z p be the ring of p adic integers. We denote the following tensor products by B p (G) =Z p B(G) and Bp (G) =Z p B(G), Z Z where we have that B p (G) isaz p order, being B p (G) its maximal order. For further information about orders, see [3]. Definition 1.3. We define the zeta function ζ Λ (s) of an order Λ, as: ζ Λ (s) := Λ I s, I Λ, ideal (Λ : I) < which, in cases B(G), B(G), Bp (G) and B p (G), converges uniformly on compact subsets of {s C Re(s) > 1}. Let Λ and Λ i be orders, for i =1,..., n and let Λ p := Z p ZΛ, which is an order over Z p. We see that the function ζ satisfies the following properties: i).- If Λ = n i=1 Λ i, we have ζ Λ (s) = n i=1 ζ Λ i (s). ii).- ζ Λ (s) = p prime ζ Λ p (s), the Euler product. For further information about Solomon s zeta function, See [4]. 2. Soluble components of B p (G). Notation. Given a subgroup H G and p Z a rational prime, we denote by O p (H) the smallest normal subgroup K H, such that H/K is a p group. Wy say that a group H is p perfect if O p (H) =H. We observe that if p H, then H is a p perfect group. Let N G (H) be the normalizer of H in G, we denote

1210 D. Villa-Hernández by W G (H) =N G (H)/H the Weyl group of H in G. Finally, we denote by e p G,H the primitive idempotents of B(G). According to the previous notation, we know that B p (G) = B p (G)e p G,H H C(G) O p (H) =H and we obtain from [7, theorem 3.1] when Π = {p}, the following relation between the Burnside Ring B p (G) and its soluble components (1) B p (G) = B p (W G (H))e p W G (H),1 H C(G) O p (H) =H where we also have that, for each soluble component (2) B p (W G (H))e p W G (H),1 = T C(W G (H)) T p -group Z p (W G (H)/T ). In the following, we will use these expressions to compute B p (G). Proposition 2.1. Let G be a group such that p G, this means that p divides the order of G but not p 2. Then, we have: B p (G)e p G,1 = B p (C p ). Proof. Let P be the Sylow p subgroup of G. From (2) we have that B p (G)e p G,1 = Z p (G/ 1 ) Z p (G/P ), where (G/ 1 )(G/ 1 ) = G (G/ 1 ). We denote: a = p (G/ 1 ), G whence a 2 = pa. Furthermore, we observe that 1 (G/ 1 ), (G/P ). Let λ 1,λ 2 Z p such that 1=λ 1 (G/ 1 )+λ 2 (G/P ), if we apply ϕ P, we obtain that 1=λ 2 ϕ P (G/P ), then, we have 1 λ 2 = W (P ).

Zeta functions of Burnside rings 1211 Hence, it is easy to see that Therefore, (G/P )= W (P ) λ 1 W (P ) G p a. B p (G)e p G,1 = Z p Zp a. We observe that B p (C p )=Z p (C p C p ) Z p (C p pc p ), where C p C p = 1 and (C p pc p )(C p pc p )=p(c p pc p ). For further information about the structure of B p (C p ) see [6, sections 2,3]. Finally, we observe the following isomorphism of rings B p (C p ) = Bp (G)e p G,1 C p C p 1 C p pc p a which proves this proposition. 3. The function ζ B(S3 ) (s). Let S 3 be the symmetric group of order 6. The conjugacy classes of the subgroups of S 3 are: C (S 3 )={1; (12) = C 2 ; (123) = C 3 ; S 3 }, where we have: C 2 S 3 1 C 3 S 3. Furthermore, we have: O 2 (1) = 1, O 3 (1) = 1, W S3 (1) = S 3 /1 = S 3 O 2 (C 2 )=1, O 3 (C 2 )=C 2, W S3 (C 2 )=C 2 /C 2 = 1 O 2 (C 3 )=C 3, O 3 (C 3 )=1, W S3 (C 3 )=S 3 /C 3 = C2 O 2 (S 3 )=C 3, O 3 (S 3 )=S 3, W S3 (S 3 )=S 3 /S 3 = 1. From (1) and the previous chart, we have that B 2 (S 3 )=B 2 (S 3 )e 2 s 3,1 B2 (C 2 )e 2 C 2,1,

1212 D. Villa-Hernández hence, from proposition 2.1 we obtain B 2 (S 3 )=[B 2 (C 2 )] 2. Apart from the result obtained in [6, section 3] for p =2, we have that (3) ζ B2(S 3) (s) = [ 1 2 s +2 1 2s]2 ζ 4 Z 2 (s), for which, we observe that the following relation is fulfilled ζ B2 (S 3 ) (s) ζ B2 (S 3 ) (1 s) = [ 2 1 2s] 2 ζz 4 2 (s) ζz 4 2 (1 s). On the other hand, from (1) and the previous chart, we have that B 3 (S 3 )=B 3 (S 3 )e 3 s 3,1 B3 (1)e 3 1,1] 2, hence, from (2) and proposition 2.1, we obtain that B 3 (S 3 )=B 3 (C 3 ) Z 2 3. Apart from the result obtained in [6, section 3] for p =3, we have that (4) ζ B3(S 3) (s) = [ 1 3 s +3 1 2s] ζ 4 Z 3 (s), for which we observe that the following relation is fulfilled ζ B3 (S 3 ) (s) ζ B3 (S 3 ) (1 s) = [ 3 1 2s] ζ Z 4 3 (s) ζz 4 3 (1 s). Finally, from the Euler product along with (3) and (4), we have that ζ B(S3) (s) = [ 1 2 s +2 1 2s] 2 [ 1 3 s +3 1 2s] ζ 4 Z (s), for which the following relation is fulfilled

Zeta functions of Burnside rings 1213 ζ B(S3 ) (s) ζ B(S3 ) (1 s) = [ 2 1 2s] 2 [ ] 3 1 2s ζz 4 (s) ζz 4 (1 s). 4. The function ζ B(A4 ) (s). Let A 4 be the alternating group of order 12. From [5] we have that the conjugacy classes of the subgroups of A 4 are: C (A 4 )= {1; (12) (34) = C 2 ; (243) = C 3 ; (13) (24) ; (12) (34) = V ; A 4 }, where we have Furthermore: C 2 V A 4 1 C 3 A 4. O 2 (1) = 1, O 3 (1) = 1, W A4 (1) = A 4 /1 = A 4 O 2 (C 2 )=1, O 3 (C 2 )=C 2, W A4 (C 2 )=V/C 2 = C2 O 2 (C 3 )=C 3, O 3 (C 3 )=1, W A4 (C 3 )=C 3 /C 3 = 1 O 2 (V ) = 1, O 3 (V )=V, W A4 (V ) = A 4 /V = C 3 O 2 (A 4 )=A 4, O 3 (A 4 )=V, W A4 (A 4 )=A 4 /A 4 = 1. From (1) and the previous chart, we have B 2 (A 4 )=B 2 (A 4 )e 2 A 4,1 B2 (1)e1,1] 2 2, hence, from (2) we obtain B 2 (A 4 )=B 2 (A 4 )e 2 A 4,1 [Z2 ] 2. Furthermore, B 2 (A 4 )e 2 A 4,1 = Z 2 (A 4 /1) Z 2 (A 4 /C 2 ) Z 2 (A 4 /V ), for which the marks regarding the ordered basis {(A 4 /1) ; (A 4 /C 2 );(A 4 /V )} is

1214 D. Villa-Hernández ϕ 1(A 4 /1) ϕ 1 (A 4 /C 2 ) ϕ 1 (A 4 /V ) ϕ C2 (A 4 /1) ϕ C2 (A 4 /C 2 ) ϕ C2 (A 4 /V ) = 12 6 3 0 2 3. ϕ V (A 4 /1) ϕ V (A 4 /C 2 ) ϕ V (A 4 /V ) 0 0 3 If we apply elementary column operations using elements in Z 2 we observe that this is equivalent to 4 2 1 0 2 1, 0 0 1 which corresponds to the marks of B 2 (C 4 ), hence, B 2 (A 4 )=B 2 (C 4 ) Z 2 ] 2. From [6, section 4] for p =2, we obtain (5) ζ B2(A 4) (s) = [1 (2) 1 s + 7 (2) 2s (2) 2 3s +3(2) 1 4s + (2) 2 5s ] ζ 5 Z 2 (s). On the other hand, from (1) and the previous chart, we have that B 3 (A 4 )=B 3 (A 4 )e 3 A 4,1 B3 (C 2 )e 3 C 2,1 B3 (C 3 )e 3 C 3,1. Hence, from proposition 2.1 and (2), we have that B 3 (A 4 )=B 3 (C 3 ) Z 3 B3 (C 3 ). Apart from the result obtained in [6, section 3] for p =3, we have that (6) ζ B3(A 4) (s) = [ 1 3 s +3 1 2s] 2 ζ 5 Z3 (s), for which the following relation is fulfilled: ζ B3 (A 4 ) (s) ζ B3 (A 4 ) (1 s) = [ 3 1 2s] 2 ζz 5 3 (s) ζz 5 3 (1 s). Finally, from the Euler product along with (5) and (6), we obtain

Zeta functions of Burnside rings 1215 ζ B(S3) (s) =f C4 ( 2 s )[ f C3 ( 3 s )] 2 ζ 5 Z (s), where, f C4 (2 s )= [ 1 (2) 1 s +7(2) 2s (2) 2 3s +3(2) 1 4s + (2) 2 5s] and f C3 (3 s )=[1 3 s +3 1 2s ]. 5. The function ζ B(A5 ) (s). Let A 5 be the alternating group of order 60. From [5] we have that the conjugacy classes of the subgroups of A 5 are: C (A 5 )={1; C 2 ; C 3 ; C 5 ; V ; S 3 ; D 10 ; A 4 ; A 5 } where: C 2 = (23) (45), C 3 = (345), C 5 = (12345), V = (23) (45) ; (24) (35), S 3 = (345) ; (12) (45) and D 10 = (12345) ; (25) (34), for which we have: 1 C 2 S 3 A 5 V A 4 A 5 D 10 A 5 C 3 S 3 A 5 C 5 D 10 A 5.

1216 D. Villa-Hernández Furthermore, we have: O 2 (1 ) = 1, O 3 (1 ) = 1, O 5 (1 ) = 1, W A5 (1 ) = A 5 O 2 (C 2 )=1, O 3 (C 2 )=C 2 O 5 (C 2 )=C 2 W A5 (C 2 )=C 2 O 2 (C 3 )=C 3, O 3 (C 3 )=1, O 5 (C 3 )=C 3 W A5 (C 3 )=C 2 O 2 (C 5 )=C 5, O 3 (C 5 )=C 5 O 5 (C 5 )=1, W A5 (C 5 )=C 2 O 2 (S 3 )=C 3, O 3 (S 3 )=S 3 O 5 (S 3 )=S 3 W A5 (S 3 )=1 O 2 (V )=1, O 3 (V )=V O 5 (V )=V W A5 (V )=C 3 O 2 (D 10 )=C 5, O 3 (D 10 )=D 10 O 5 (D 10 )=D 10 W A5 (D 10 )=1, O 2 (A 4 )=A 4, O 3 (A 4 )=V O 5 (A 4 )=A 4 W A5 (A 4 )=1, O 2 (A 5 )=A 5, O 3 (A 5 )=A 5 O 5 (A 5 )=A 5 W A5 (A 5 )=1. From (1) and the previous chart, we have that B 2 (A 5 )=B 2 (A 5 )e 2 A 5,1 B2 (C 2 )e 2 C 2,1] 2 B2 (1)e 2 1,1] 2, hence, from 2.1 and (2), we have that Furthermore, B 2 (A 5 )=B 2 (A 5 )e 2 A 5,1 [B2 (C 2 )] 2 Z 2 ] 2. B 2 (A 5 )e 2 A 5,1 = Z 2 (A 5 /1) Z 2 (A 5 /C 2 ) Z 2 (A 5 /V ), for which the marks regarding the ordered basis {(A 5 /1) ; (A 5 /C 2 );(A 5 /V )} is ϕ 1(A 5 /1) ϕ 1 (A 5 /C 2 ) ϕ 1 (A 5 /V ) 60 30 15 ϕ C2 (A 5 /1) ϕ C2 (A 5 /C 2 ) ϕ C2 (A 5 /V ) = 0 2 3. ϕ V (A 5 /1) ϕ V (A 5 /C 2 ) ϕ V (A 5 /V ) 0 0 3 If we apply elementary column operations using elements in Z 2 we observe that this is equivalent to 4 2 1 0 2 1, 0 0 1 which corresponds to the marks of B 2 (C 4 ), hence, B 2 (A 5 )=B 2 (C 4 ) B 2 (C 2 )] 2 Z 2 ] 2.

Zeta functions of Burnside rings 1217 From [6, sections 3,4] for p =2, we obtain: (7) ζ B2(A 5) (s) = [ 1 (2) 1 s + 7 (2) 2s (2) 2 3s +3(2) 1 4s + (2) 2 5s ] [1 2 s +2 1 2s] 2 ζ 9 Z2 (s). On the other hand, from (1) and the previous chart, we have that B 3 (A 5 )=B 3 (A 5 )e 3 A 5,1 B3 (C 2 )e 3 2 C 2,1] B3 (C 3 )e 3 C 3,1 B3 (1)e1,1] 3 3, hence, from proposition 2.1 and (2), we obtain B 3 (A 5 )=[B 3 (C 3 )] 2 Z 5 3. Apart from the result obtained in [6, section 3] for p =3, we have that (8) ζ B3(A 5) (s) = [ 1 3 s +3 1 2s]2 ζ 9 Z 3 (s), for which the following relation is fulfilled: ζ B3 (A 5 ) (s) ζ B3 (A 5 ) (1 s) = [ 3 1 2s] 2 ζz 9 3 (s) ζz 9 3 (1 s). In the same way, from (1) and the previous chart, we have B 5 (A 5 )=B 5 (A 5 )e 5 A 5,1 B5 (C 2 )e 5 C 2,1] 2 B5 (C 3 )e 5 C 3,1 B5 (1)e 5 1,1 hence, from proposition 2.1 and (2), we obtain B 5 (A 5 )=B 5 (C 5 ) Z 7 5. Apart from the result obtained in [6, section 3] for p =5, we have that ] 4 (9) ζ B5(A 5) (s) = [ 1 5 s +5 1 2s] ζ 9 Z 5 (s) for which the following relation is fulfilled: ζ B5 (A 5 ) (s) ζ B5 (A 5 ) (1 s) = [ 5 1 2s] ζ Z 9 5 (s) ζz 9 5 (1 s).

1218 D. Villa-Hernández Finally, from the Euler product along with (7), (8), and (9) we have that ζ B(A5) (s) =f C4 ( 2 s ) f 2 C 2 ( 2 s ) f 2 C 3 ( 3 s ) f C5 ( 5 s ) ζ 9 Z (s), where: f C4 (2 s )= [ 1 (2) 1 s +7(2) 2s (2) 2 3s +3(2) 1 4s + (2) 2 5s], f C2 (2 s )=[1 2 s +2 1 2s ], f C3 (3 s )=[1 3 s +3 1 2s ] and f C5 (5 s )=[1 5 s +5 1 2s ]. 6. The function ζ B3 (S 4 ) (s). Let S 4 be the symmetric group of order 24. From [5] we have that the conjugacy classes of the subgroups of S 4 are: C (S 4 )={1; C 2 ; C 2; C 3 ;(C 2 C 2 ); C 4 ; V ; S 3 ; D 8 ; A 4 ; S 4 } where: C 2 = (34) = (13) (14), C 2 = (13) (24) = (12) (34) (14), C 3 = (243), C 2 C 2 = (12) ; (34) = (24) ; (13) (14), C 4 = (1324) = (1432) (14), V = (14) (23) ; (13) (24), S 3 = (34) ; (243), D 8 = (1234) ; (13) and

Zeta functions of Burnside rings 1219 A 4 = (13) (24) ; (14) (23) ; (243), with the following contentions: C 2 C 2 D 8 S 4 C 2 1 C 2 C 3 S 3 S 4 where C 2 D 8 and V S 4, hence: C 2 C 2 D 8 S 4 C 4 D 8 S 4 D 8 S 4 V A 4 S 4 S 3 S 4 A 4 S 4 O 2 (1 ) = 1 O 3 (1 ) = 1 W S4 (1 ) = S 4 O 2 (C 2 )=1 O 3 (C 2 )=C 2 W S4 (C 2 )=C 2 O 2 (C 2 )=1 O 3 (C 2 )=C 2 W S4 (C 2 )=C 2 C 2 O 2 (C 3 )=C 3 O 3 (C 3 )=1 W S4 (C 3 )=C 2 O 2 (C 2 C 2 )=1 O 3 (C 2 C 2 )=C 2 C 2 W S4 (C 2 C 2 )=C 2 O 2 (C 4 )=1 O 3 (C 4 )=C 4 W S4 (C 4 )=C 2 O 2 (V )=1 O 3 (V )=V W S4 (V )=S 3 O 2 (D 8 )=1 O 3 (D 8 )=D 8 W S4 (D 8 )=1 O 2 (S 3 )=C 3 O 3 (S 3 )=S 3 W S4 (S 3 )=1 O 2 (A 4 )=A 4 O 3 (A 4 )=V W S4 (A 4 )=C 2 O 2 (S 4 )=A 4 O 3 (S 4 )=S 4 W S4 (S 4 )=1. From (1) and the previous chart, we obtain: B 3 (S 4 )= H B 3 (H)e 3 H,1 B3 (1)e 3 1,1] 3 B3 (C 2 )e 3 C 2,1] 2,

1220 D. Villa-Hernández where H goes over the set {S 4 ;(C 2 C 2 ); C 2 ; S 3}. Hence, from proposition 2.1 and (2), we obtain B 3 (S 4 )=[B 3 (C 3 )] 2 Z 7 3. Apart from the result obtained in [6, section 3] for p =3, we have that (10) ζ B3(S 4) (s) = [ 1 3 s +3 1 2s]2 ζ 11 Z 3 (s), for which following relation is fulfilled: ζ B3 (S 4 ) (s) ζ B3 (S 4 ) (1 s) = [ 3 1 2s] 2 ζz 11 3 (s) ζz 11 3 (1 s). Acknowledgment. I am grateful to Prof. Dr. Alberto Gerardo Raggi Cárdenas for his many valuable suggestions. References [1] Bouc, S. Burnside rings. (2000). Handbook of algebra, vol 2, 739-804, North-Holland, Amsterdam. [2] 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. [3] Reiner, I.(1975). Maximal Orders, London-New York. Academic Press. [4] Solomon, L. (1977). Zeta Functions and Integral Representation Theory. Advances in Mathematics 26, 306-326. [5] The GAP Group. (2008) GAP Groups, Algorithms, and Programming, Version 4.4.12. (http://www.gap-system.org) [6] Villa-Hernandez D. (2011). Zeta Function of the Burnside Ring for Cyclic Groups. International Journal of Algebra, Vol. 5, Number 26, pp 1255-1266. [7] 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: June, 2012