The class number of an abelian group

Abstract

The groups in this paper are abelian. Let G be a reduced torsion-free finite rank group. Then G is cocommutative if End ( G ) is commutative modulo the nil radical. The class number of G, h ( G ) , is the number of isomorphism classes of groups H that are locally isomorphic ( = nearly isomorphic) to G. We say that G satisfies the power cancellation property if G n ≅ H n for some group H and integer n > 0 implies that G ≅ H . We say that G has a Σ-unique decomposition if G n has a unique direct sum decomposition for each integer n > 0 . Let P o ( G ) = { groups H | H ⊕ H ′ = G m for some group H ′ and some integer m > 0 } . We say that G has internal cancellation if given H , K , L ∈ P o ( G ) such that H ⊕ K ≅ H ⊕ L then K ≅ L . We use the class number to study the torsion-free finite rank groups G that have the power cancellation property, or a Σ-unique decompositions, or the internal cancellation property. Furthermore, we show that the power cancellation property for cocommutative strongly indecomposable reduced torsion-free finite rank groups is equivalent to the problem of determining the class number of an algebraic number field. Let G ¯ be the integral closure of G. Using the Mayer–Vietoris sequence we show that there are finite groups associated with G of orders L ( p ) , m ˆ p , and n ˆ p such that h ( G ) = h ( G ¯ ) m ˆ p L ( p ) n ˆ p . Let E ( p ) = Z + p E ¯ where p is a rational prime and E ¯ is the ring of algebraic integers in the algebraic number field k = Q E ¯ . Let G ( p ) be a group such that End ( G ( p ) ) = E ( p ) . We show that the sequence { L ( p ) h ( G ( p ) ) / h ( G ¯ ( p ) ) | primes p } is asymptotically equal to the sequence { p f − 1 | primes p } where f = [ k : Q ] . Furthermore, for quadratic number fields k, h ( k ) = 1 iff { L ( p ) h ( G ( p ) ) | primes p } is asymptotically equal to the sequence of rational primes. This connects unique factorization in number fields with the sequence of rational primes, and with direct sum properties of integrally closed cocommutative strongly indecomposable torsion-free finite rank groups.