Algebra and Logic最新文献

There is a well-known factorization of the number 2^2m + 1, with m odd, related to the orders of tori of simple Suzuki groups: 2^2m +1 is a product of a = 2^m + 2^(m+1)/2 +1 and b = 2^m − 2^(m+1)/2 + 1. By the Bang–Zsigmondy theorem, there is a primitive prime divisor of 2^4m − 1, that is, a prime r that divides 2^4m − 1 and does not divide 2ⁱ − 1 for any 1 ≤ i < 4m. It is easy to see that r divides 2^2m + 1, and so it divides one of the numbers a and b. It is proved that for every m > 5, each of a, b is divisible by some primitive prime divisor of 2^4m − 1. Similar results are obtained for primitive prime divisors related to the simple Ree groups. As an application, we find the independence and 2-independence numbers of the prime graphs of almost simple Suzuki–Ree groups.

A group G is said to be m-rigid if it contains a normal series of the form G = G₁ > G₂ > . . . > G_m > G_m+1 = 1, whose quotients G_i/G_i+1 are Abelian and, treated as (right) ℤ[G/G_i]-modules, are torsion-free. A rigid group G is said to be divisible if elements of the quotient ρ_i(G)/ρ_i+1(G) are divisible by nonzero elements of the ring ℤ[G/ρ_i(G)]. Previously, it was proved that the theory of divisible m-rigid groups is complete and ω-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible m-rigid group G.

We give a description of finite 4-primary groups with disconnected Gruenberg–Kegel graph containing a triangle. As a corollary, finite groups whose Gruenberg–Kegel graph coincides with the Gruenberg–Kegel graph of ³D₄(2) are exemplified, which generalizes V. D. Mazurov’ description of finite groups isospectral to the group ³D₄(2).

A group G is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups H in the factor group N_G(H)/H, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group G which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution z with the property that the centralizer C_G(z) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group G that is saturated with finite almost simple groups and contains an involution z with the property that the centralizer C_G(z) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.

For a finite group G, the spectrum is the set ω(G) of element orders of the group G. The spectrum of G is closed under divisibility and is therefore uniquely determined by the set μ(G) consisting of elements of ω(G) that are maximal with respect to divisibility. We prove that a finite group isospectral to Aut(J₂) is unsolvable.

Families 𝒫_I consisting of permutations of the natural numbers ω whose degrees belong to an ideal I of Turing degrees, as well as their jumps ({mathcal{P}}_{mathrm{I}}{prime}), are studied. For any countable Turing ideal I, the degree spectra of families 𝒫_I and their jumps ({mathcal{P}}_{mathrm{I}}{prime}) are described. For some ideals I generated by c.e. degrees, the spectra of families 𝒫_I are defined.

Previously, Došen and Božić introduced four independent intuitionistic modal logics, one for each of four types of modal operators—necessity N, possibility P, impossibility Im, and unnecessity Un. These logics are denoted HKM, where M ∈ {N, P, Un, Im}. Interest in treating the four types of modal operators separately is associated with just the fact that these cannot be reduced to each other over intuitionistic logic. Here we study extensions of logics HKM that have normal companions. It turns out that all extensions of the logics HKN and HKUn possess normal companions. For the extensions of HKP and HKIm, we obtain a criterion for the existence of normal companions, which is postulated as the presence of some modal law of double negation. Also we show how adding of this law influences expressive capacities of a logic. Of particular interest is the result saying that extensions of HKP and HKIm have normal companions only if they are definitionally equivalent to those of HKN and HKUn respectively. This result is one more example of the differences in behavior of the four types of modal operators over intuitionistic logic.

Generic algorithms decide problems on sets of almost all inputs, outputting an indefinite answer for other rare inputs. We will prove that the word problem is generically decidable in finitely generated semigroups 𝔖, for which there exists a congruence θ such that the semigroup 𝔖/θ is an infinite residually finite monoid with cancellation property and decidable word problem. This generalizes the author’ earlier result on generic decidability of the word problem in finitely presented semigroups that remain infinite when adding commutativity and cancelling properties. Examples of such semigroups are one-relator semigroups as well as so-called balanced semigroups, for which generic decidability of the word problem has been proved by Won. In particular, balanced are Tseitin and Makanin’s classical semigroups with undecidable word problem.