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We study a group G containing an element g such that C G (g)∩g G is finite. The nonoriented graph Γ is defined as follows. The vertex set of Γ is the conjugacy class g G . Vertices x and y of the graph G are bridged by an edge iff x≠y and xy=yx. Let Γ 0 be some connected component of G. On...
For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp.,...
Using the group , we refute the conjecture dubbed in 1976 by V. Belyaev and N. Sesekin, which maintained that the growth function σ(n) of a finitely generated group satisfies the inequality σ(n)≤(σ(n−1)+σ(n+1))/2 for all sufficiently large n.
We give a description of simple nonassociative (−1,1)-superalgebras of characteristic ≠ 2, 3. It is proved that in such a superalgebra B, the even part A is a differentially simple, associative, and commutative algebra and the odd part M is a finitely generated, associative, and commutative...
It is proved that a lattice of quasivarieties of an arbitrary variety of commutative Moufang loops either has the power of the continuum or is finite, and that the latter is the case iff is generated by a finite group. It is also stated that the lattice of all quasivarieties of a least...
The structure of scalar fields for a directly indecomposable finite-dimensional algebra treated as a ring is studied. Scalar fields are assumed similar if their action on a ring is identical modulo an annihilator. The criterion for a class of maximal scalar fields to be unique under a similitude...
Part of any basis of a relatively free group in the variety is called a primitive system of elements. We provide a criterion of being primitive for , where is a variety of Abelian groups satisfying x m =1, and a variety generated by a finite group. Let be a variety of nilpotent groups of class...
For an n-ary Boolean function β, we introduce the concept of a weakly β-combinatorial selector set and denote a class of all β-combinatorial selector sets by R(β). A description of the classes R(β) is given—in particular, if β is a nonmonotone Boolean function then R(β) coincides either...
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