Volume 25, Issue 4

Let G be a finite group and f be an automorphism of the group G. The automorphism f specifies a recurrent sequence {a i } on the group G, i = 0, 1, . . ., according to the rule a i+1 = f(a i ). If a 0 is the initial element of the sequence, then its period does not exceed the number of elements in the group having the same order as the element a 0 . Thus, it makes sense to formulate the question of whether there exist groups in which such recurrent sequence for a certain automorphism has the maximal period for any initial element. In this paper we introduce the notion of an automorphism of the maximal period and find all Abelian groups and finite groups of odd orders having automorphisms of the maximal period. Also, a number of results for finite groups of even orders are established.

A classification of correlation-immune and minimal corelation-immune Boolean function of 4 and 5 variables with respect to the Jevons group is given. Representatives of the equivalence classes of correlationimmune functions of 4 and 5 variables are decomposed into minimal correlation-immune functions. Characteristics of various decompositions of the constant function 1 into minimal correlation-immune functions are presented.

The solvability of the equational theory of commutative medial n-ary groupoids is in the paper used to describe the structure of the free commutative medial n-ary groupoid and to demonstrate that a commutative medial n-ary groupoid has a convex linear representation.

The paper is concerned with circuits in the basis {x →y ,x̅}. The exact value of the complexity of implementation of an even parity function is obtained and the minimal circuits implementing an odd parity function are described

All 23 closed classes of three-valued logic that contain essentially multiplace linear functions are defined. It is shown that the order of every such class is not greater than 3.

For an arbitrary prime power q, a criterion for irreducibility of a polynomial of the form over the field K = GF(qt) is established.

An invariant ring A is arithmetical if and only if every finitely generated ideal M of the ring A is a quasi-projective A-module and every endomorphism of this module may be extended to an endomorphism of the module A A . An invariant semiprime ring A is arithmetical if and only if every finitely generated ideal M of the ring A is a quasi-projective A-module.