Revistes Catalanes amb Accés Obert (RACO)

On the product of two π-decomposable soluble groups

L. S. Kazarin, A. Martínez Pastor, M. D. Pérez-Ramos


Let the group G = AB be a product of two π-decomposable sub-groups A = Oπ(A) × Oπ′ (A) and B = Oπ(B) × Oπ′ (B) where π is a set of primes. The authors conjecture that Oπ(A)Oπ(B) = Oπ(B)Oπ(A) if π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if Oπ ′(A) and Oπ′(B) have coprime orders.

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