A condition of indecomposability for Butler $$\mathrm B (n)$$ B (
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A condition of indecomposability for Butler B(n)-groups Clorinda De Vivo · Claudia Metelli
© Akadémiai Kiadó, Budapest, Hungary 2014
Abstract We give a manageable sufficient condition for indecomposability of Butler B(n)groups, allowing the easy construction of a big family of indecomposable torsionfree Abelian groups of finite rank. Keywords Abelian group · Torsionfree · Finite rank · Butler group · B(n)-group · Type · Tent Mathematics Subject Classification
20K15 · 06F99 · 06B99
1 Introduction In this paper group = torsionfree Abelian group of finite rank; greek letters denote rational numbers; J = {1, . . . , r }. A B(n)-group W is the sum of a finite number r of pure rank one subgroups, Wr =∗ + · · · + ∗ , subject to a system S of n ≤ r independent linear relations αl, j w j = 0 (l = 1, . . . , n); equivalently, it is a torsionfree quotient j=1
W = Y /K Y of a completely decomposable (= c.d.) group Y = ∗ ⊕ · · · ⊕ ∗ over a rank n pure subgroup K Y , collecting the linear relations among the w j ’s. Thus a B(n)-group is determined by two choices: an order-theoretical one (the isomorphism types u j = tW (w j ) of the rank 1 subgroups ∗ , called base types of W ) and a linear one (the relations in K Y ).
C. De Vivo Via Coriolano 21, 80124 Napoli, Italy e-mail: [email protected] C. Metelli (B) Via Tre Garofani 41, 35124 Padova, Italy e-mail: [email protected]
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C. De Vivo, C. Metelli
As customary in this subject, we use quasi-isomorphisms ( = isomorphisms up to finite index, [5]), instead of isomorphisms, as basic equivalence; and write “isomorphic, direct decomposition, pure subgroup, ...” instead of “quasi-isomorphic, quasi-direct decomposition, quasi-pure subgroup, ...” . In his defining paper of 1967 [2], M. Butler proved in particular that the typeset of a Butler group W , i.e. the set of (isomorphism) types of the pure rank 1 subgroups of W , is a finite sub-∧-semilattice of the lattice T(∧, ∨) of all types. As usual, we represent the base types u j of W as products of Primes (capitalized, to avoid confusion with the prime numbers of Q; see Sect. 2), thus giving the order-theoretical part of the definition in the form of a table called tent (Example 2.1). In the amply studied class B(1) [1], where the single relation may be assumed to be the diagonal relation w1 + · · · + wr = 0, the necessary and sufficient condition for decomposability is purely order-theoretical [6]. In general, though, this is not true: in [3, Example 4.8] we have two B(2)-groups with the same base types (the same tent), only one of which is decomposable; in fact, even if the tent is the same, the typeset needn’t be: it depends also on the coefficients αl, j of the relations. In this paper we give an order-theoretical sufficient condition for indecomposability of B(n)-groups; it insures indecomposability if the tent is ‘complicated enough’, that is, if there are ‘enough Primes’. The way to make this statement precise delves into the deep nature of Primes [4], which itself, curiously, is essentially linear. The condition
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