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There exist various connections between homomorphism groups and direct sums and products of abelian groups. For instance, natural isomorphisms Hom(A,∏Bi ) ≅ ∏Hom(A,Bi) and Hom( ⊕Bi, A) = ∏Hom (Bi,A) are often used. In this paper, as well as in [1], we consider the case when Hom (A,⊕Bi)= Hom (A,∏Bi), which is equivalent to existence of the natural isomorphism Hom (A, ⊕Bi) = ∏Hom (A,Bi). Let M be a set of groups {Z(p) | p runs through an infinite set T of prime numbers}. An abelian group is called M-large if Hom(A, ⊕ Z(p)) = Hom( А,∏z(р)). This paper presents characterization of torsion free and mixed M -large groups.

In the paper, some new results on bent sum decomposition problem are discussed. It is proved that any Boolean function in n variables of degree d ^ n/2 can be represented as the sum of not more than 21 ) bent functions, A where b ^ d and b is the least integer such that 2b|n.

In this paper we study the existence of bent functions on the minimal distance to given bent functions and their regular properties.

It is known that a definition of Boolean bent functions is invariant under any linear nonsingular transformation of the variables. This paper investigates the effect of the basis selection of the Boolean function reduced representation on its property to be a hyper-bent function. The following results are obtained: 1) for any bent function of 4 variables there exist two bases of the vector space (F24 )f such that the reduced representation of this function in the first basis is a hyper-bent function, and in the second basis is not. 2) For any even n > 4 there exist two bases of the vector space (F2n )F and the function of n variables such that the reduced representation of this function in the first basis is a hyper-bent function, and in the second basis is not.