Algebra and Logic最新文献

Let G be a finite group of Lie type, and T some maximal torus of the group G. We bring to a close the study of the question of whether there exists a complement for a torus T in its algebraic normalizer N (G, T). It is proved that any maximal torus of a group G ∈ {G₂(q), ²G₂(q), ³D₄(q)} has a complement in its algebraic normalizer. Also we consider the remaining twisted classical groups ²A_n(q) and ²D_n(q).

Let π be a proper subset of the set of all prime numbers. Denote by r the least prime number not in π, and put m = r, if r = 2, 3, and m = r − 1 if r ≥ 5. We look at the conjecture that a conjugacy class D in a finite group G generates a π-subgroup in G (or, equivalently, is contained in the π-radical) iff any m elements from D generate a π-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type ²B₂(q), ²G₂(q), G₂(q), and ³D₄(q).

Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer C_G(A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of (|A|,m)-bounded index which has Fitting height at most 2α(A) + 2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer C_G(A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of (|A|, r)-bounded index which has Fitting height at most 4α(A) + 4α(A) + 3. Here, a left Engel sink of an element g of a group G is a set 𝔈 (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g], . . . , g] belong to 𝔈 (g). (Thus, g is a left Engel element precisely when we can choose (g) = {1}.) A right Engel sink of an element g of a group G is a set ℜ(g) such that for every x ∈ G all sufficiently long commutators [...[[g, x], x], . . . , x] belong to ℜ(g). Thus, g is a right Engel element precisely when we can choose ℜ(g) = {1}.

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.