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In this paper, building among others on earlier works by U. Krause and C.Zahlten (dealing with the case of cyclic groups), we obtain a new upper boundfor the little cross number valid in the general case of arbitrary finiteAbelian groups. Given a finite Abelian group, this upper bound appears todepend only on the rank and on the number of distinct prime divisors of theexponent. The main theorem of this paper allows us, among other consequences,to prove that a classical conjecture concerning the cross and little crossnumbers of finite Abelian groups holds asymptotically in at least two differentdirections.

A new upper bound for the cross number of finite Abelian groups