Algebra and Logic最新文献

We study periodic groups saturated with finite simple symplectic groups.

For a finite group G, we denote by N (G) the set of its conjugacy class sizes. Recently, the following question was posed: given any n ∈ ℕ and an arbitrary non-Abelian finite simple group S, is it true that G ≃ Sⁿ if G is a group with trivial center and N (G) = N (Sⁿ)? The answer to this question is known for all simple groups S with n = 1, and also for S ∈ {A₅, A₆}, where A_k denotes the alternating group of degree k, with n = 2. It is proved that the group A₅ × A₅ × A₅ is uniquely defined by the set N (A₅ × A₅ × A₅) in the class of finite groups with trivial center.

We study degrees and degree spectra of groups ({mathfrak{G}}_{mathrm{I}}) defined on a set of permutations of the natural numbers ω whose degrees belong to a Turing ideal I. A necessary condition and a sufficient condition are stated which specify whether an arbitrary Turing degree belongs to the degree spectrum of a group ({mathfrak{G}}_{mathrm{I}}). Nonprincipal ideals I for which the group ({mathfrak{G}}_{mathrm{I}}) has or does not have a degree are exemplified.

A variety of associative algebras is nonmatrix if it does not contain the algebra of 2 × 2 matrices over a given field. Nonmatrix varieties were introduced and studied by V. N. Latyshev in [Algebra and Logic, 16, No. 2, 98-122 (1977); Algebra and Logic, 16, No. 2, 122-133 (1977); Mat. Zam., 27, No. 1, 147-156 (1980)] in connection with the Specht problem. A series of equivalent characterizations of nonmatrix varieties was obtained in [Isr. J. Math., 181, No. 1, 337-348 (2011)]. In the present paper, the notion of nonmatrix variety is extended to nonassociative algebras, and their characterization from the last-mentioned paper is generalized to alternative, Jordan, and some other varieties of algebras.

We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the π-radical of a finite group (here π is an arbitrary set of primes). For an odd prime number r, put m = r, if r = 3, and m = r - 1 if r ≥ 5. Let L be a simple non-Abelian group whose order has a prime divisor s such that s = r if r divides |L| and s > r otherwise. Suppose also that x is an automorphism of prime order of L. Then some m conjugates of x in the group (langle L,xrangle ) generate a subgroup of order divisible by s. The conjecture is confirmed for the case where L is a group of Lie type and x is an automorphism induced by a unipotent element.

We introduce a bipolar classification with index j for endomorphisms of an arbitrary n-groupoid with n > 1, where j = 1, 2, . . . , n. The classifications of endomorphisms constructed generalize the bipolar classification of endomorphisms of an arbitrary groupoid (i.e., a 2-groupoid) introduced previously. Using a left bipolar classification of endomorphisms of an n-groupoid (a particular case of the obtained classifications), we succeed in constructing an integral classification of endomorphisms of an arbitrary algebra (i.e., a structure without relations) with finitary operations.

A finite Frobenius group in which the order of complements is divisible by a prime number p is called a Φ_p-group. We prove the theorem stating the following. Let G be a periodic group with a finite element a of prime order p > 2 saturated with Φ_p-groups. Then G = F λ H is a Frobenius group with kernel F and complement H. If G contains an involution i commuting with the element a, then H = C_G(i) and F is Abelian, and H = N_G((langle arangle )) otherwise.

On a Cartan group ({mathbb{K}}) equipped with a Carnot–Carathéodory metric d_cc, we find the exact value of a constant in the (1, q₂)-generalized triangle inequality for its Box-quasimetric. It is proved that any two points x, y ∈ ({mathbb{K}}) can be joined by a horizontal k-broken line ({L}_{x,y}^{k}), k ≤ 6; moreover, the length of such a broken line ({L}_{x,y}^{k}) does not exceed the quantity Cd_cc(x, y) for some constant C not depending on the choice of x, y ∈ ({mathbb{K}}). The value 6 here is nearly optimal.

A finitely generated group G, which acts on a tree so that all edge stabilizers are infinite cyclic groups and all vertex stabilizers are free rank 2 Abelian groups, is called a tubular group. Every tubular group is isomorphic to the fundamental group π₁(𝒢) of a suitable finite graph 𝒢 of groups. We prove a criterion for residual π-finiteness of tubular groups presented by trees of groups. Also we state a criterion for residual p-finiteness of tubular groups whose corresponding graph contains one edge outside a maximal subtree.