HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS XUEJUN GUO 1 ADEREMI KUKU 2 1 Department of Mathematics, Nanjing University Nanjing, Jiangsu 210093, The People’s Republic of China guoxj@nju.edu.cn and The Abdus Salam International Center for Theoretical Physics, Trieste, Italy 2 Institute for Advanced Study, Princeton, NJ, USA kuku@ias.edu Abstract: In this paper we study the possible torsion in even dimensional higher class groups Cl 2n (Λ) (n ≥ 1) of an order Λ in a semi-simple algebra A over a number field F with ring of integers O F . We show that for certain orders − called generalized Eichler orders − p-torsion in Cl 2n (Λ) can only occurs for primes p dividing prime ideals ℘ of O F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders. Keywords: higher class group, quaternion algebra, Eichler order, semi-simple alge- bra, hereditary order 2000 Mathematics Subject Classification: 19D50, 19F27. 1. Introduction Let F be a number field and O F the ring of integers in F . Let A be a semi-simple algebra over F and Λ an O F -order in A. The higher class groups of Λ are defined as Cl n (Λ) = ker(SK n (Λ) −→  ℘ SK n (Λ ℘ )), where ℘ runs through all maximal ideals of O F and SK n (Λ) := ker(K n (Λ) −→ K n (A)), for all integers n ≥ 0. By the Theorem 1 and 2 in [2], Cl n (Λ) are trivial for maximal orders. Later Kuku proved in [6] that Cl n (Λ) are finite for arbitrary orders. In [4], it is proved that the only p-torsion possible in Cl 2n+1 (Λ) is for those rational primes p which lie under the prime ideals of O F at which Λ is not maximal. In this paper, we prove the analogous result for even dimensional higher class groups Cl 2n (Λ) (n ≥ 1) in 1