Characterization of 3-Rewritable Finite Nilpotent Groups

Let n be an integer greater than 1. A group G is said to be “ n-rewritable” whenever for every n elements x ₁,…,x _n of G, there exist distinct permutations τ, σ on the set {1,2,…, n} such that x _τ(1) ··· x _τ(n) = x _{σ (1)} ··· x _{σ (n)}. In this article, we complete the classification of 3-rewritable finite nilpotent groups and prove that a finite nilpotent group G is 3-rewritable if and only if G has an abelian subgroup of index 2 or the derived subgroup has order < 6.     

Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|² < |G|) or G is a direct product of 2-transitive Frobenius groups.     

ℳ-Supplemented Subgroups and Their Properties

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for any maximal subgroup H ₁ of H. In this article, we investigate the influence of ?-supplementation of some primary subgroups in finite groups. Some new results about supersolvable groups and formation are obtained.     

On normal subgroups in the periodic products of S.I. Adian

A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G ₁ is an HF subgroup of the direct product G ₁ × G ₂, as well as of the free product G ₁ * G ₂ of groups G ₁ and G ₂. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Π _i∈I ⁿ G _i of an arbitrary family of groups {G _i } _i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.     

A GENERALIZATION OF COMPLETE GROUPS

Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σ _n , where σ _n is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel the results found in the author's dissertation for Lie algebras. There it is shown that only complete Lie algebras have the desired property. Thus, these results provide an example of when the theory of Lie algebras diverges from that of groups.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

On generalizations of Fermat curves over finite fields and their automorphisms

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽_q of characteristic p>2. Assume that the 𝔽_q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups C_m and C_n of orders m and n prime to p, such that both quotient curves 𝒳∕C_n and 𝒳∕C_m are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups.     

Finite groups with some given systems of Xm‐semipermutable subgroups

Let X be a nonempty subset of a group G. We call a subgroup A of G an X_m‐semipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T₁ of T there exists an element x ∈ X such that AM^x = M^xA. In this paper, we study the structure of finite groups with some given systems of X_m‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite Groups all of Whose Second Maximal Subgroups are QTI-Subgroups

A subgroup H of a finite group G is called a QTI-subgroup if C _G (x) ≤ N _G (H) for any 1 ≠ x ∈ H. In this article, the finite groups all of whose second maximal subgroup are QTI-subgroups are classified.     

Subgroups Of Direct Products Of Elementarily Free Groups

The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass of the hyperbolic limit groups. We prove that if G ₁,...,G _n are in then a subgroup Γ ⊂ G ₁ × … × G _n is of type FP _n if and only if Γ is itself, up to finite index, the direct product of at most n groups from . This provides a partial answer to a question of Sela. This work was supported in part by Franco–British Alliance project PN 05.004. The first author is also supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Research Merit Award. Received: July 2005 Accepted: April 2006     

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N _G (K) ∩ K ^x  ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P ₀ ? P ₁ ? ··· ? P _n  = P is called a maximal chain of P provided that |P _i : P _i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P ₀ ? P ₁ ? ··· ? P _i  ? ··· ? P _n  = P such that P _i  ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.     

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular, the word and membership problems. Using our recent results on “generic-case complexity”, we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem of G is linear time, uniformly with respect to the collection of all length-invariant measures on G. This results applies to many of the groups usually studied in geometric group theory: for example, all braid groups B_n, all groups of hyperbolic knots, many Coxeter groups and all Artin groups of extra-large type.     

New Characterizations of p-Supersolubility of Finite Groups

Let H be a subgroup of a finite group G. H is said to be λ-supplemented in G if G has a subgroup T such that G = HT and H ∩ T ≤ H _SE , where H _SE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the λ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersolubility of finite groups are given under the assumption that some primary subgroups are λ-supplemented. As applications, a number of previous known results are generalized.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Nearly s-Normality of Groups and Its Properties

We call a subgroup H of a group G nearly s-normal in G if there exists N ? G such that HN ? G and H ∩ N ≤ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups.     

Centrality in Rees-Sushkevich varieties

Denote by RS _n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS _n are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C _m RS _n to be the class of all semigroups S in RS _n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup. The classes C _m RS _n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C ₂ RS _n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties C _m RS _n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C _m RS _n , where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C _∞ RS _n ) is non-finitely based. Further, the variety C _∞ RS _n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based. Received October 29, 2003; accepted in final form February 11, 2007.     

Groups with Many Rewritable Products

For any integer n ≥ 2, a group G is said to have the n-rewritable property R _n if every infinite subset X of G contains n elements x ₁,…, x _n such that the product x ₁…x _n  = x _σ(1)…x _σ(n) for some permutation σ ≠ 1. We show here that if G satisfies R _n , then G has a subgroup N of finite index with a finite central subgroup A of N such that the exponent of (N/A)/Z(N/A) is finite and has size bounded by (n ? 1)!. This extends the main result in [4 Curzio , M. , Longobardi , P. , Maj , M. , Rhemtulla , A. ( 1992 ). Groups with many rewritable products . Proc. AMS. 115 ( 4 ): 931 – 934 .[Crossref], [Web of Science ®] , [Google Scholar]] which asserts that a group G is an R _n group for some integer n if and only if G has a normal subgroup F such that G/F is finite, F is an FC-group, and the exponent of F/Z(F) is finite.     

Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T_1 of T there exists an clement x∈X such that AT_i~x=T_i~xA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.