Search

AbstractLet G be an abelian group, and let F(G) be the free commutative monoid with basis G. For Ω⊂F(G), define the universal zero-sum invariant dΩ(G) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω. The invariant dΩ(G) unifies many classical zero-sum invariants. Let B(G) be the submonoid of F(G) consisting of all zero-sum sequences over G, and let A(G) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B(G), is denoted by ε. The well-known Davenport constant D(G) of the group G can be also represented as dB(G)∖{ε}(G) or dA(G)(G) in terms of the universal zero-sum invariant. Notice that A(G) is the unique minimal generating set of the monoid B(G) from the point of view of Algebra. Hence, it would be interesting to determine whether A(G) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A(G) such that dΩ(G)=D(G). Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant dΩ;Ψ(G) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant DΨ(G) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that DΨ(G)<∞ in terms of the weights set Ψ when |Ψ| is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant. [ABSTRACT FROM AUTHOR]

AbstractA subgroup X of a group G is called pronormal if X and Xg are conjugate in 〈X,Xg〉 for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyĭ and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality ℵ in which all subgroups of cardinality ℵ are pronormal. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

AbstractThe conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019]. [ABSTRACT FROM AUTHOR]

Let G be a classical group of dimension d and let a = (a 1 , ... , a d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A (a) be a generic element in the Lie algebra g (F 〈 a 〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y ′ = A (a) y over F 〈 a 〉 is G (C) . In this paper we construct a differential field extension L of F 〈 a 〉 such that the field of constants of L is C, the differential Galois group of y ′ = A (a) y over L is still the full group G (C) and A (a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A (a) . [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

In this paper, all supercharacter theories of dicyclic groups T 4 n are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar. [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]

Strong flatness and condition (P) property of right S-acts over a monoid S have been the topic of many papers on homological classification of monoids. In this paper, we continue to investigate the monoids over which every right S-act satisfying condition (P) is strongly flat and we obtain some new classes of monoids. [ABSTRACT FROM AUTHOR]

Let SVir be the well-known super-Virasoro algebras. In this paper, we first prove that any super-skewsymmetric super-biderivation of SVir is inner. Based on this, we show that every linear super-commuting map ψ on SVir is of the form ψ(x) = f(x)c, wherefis a linear function from SVir to ℂ mapping the odd part of SVir to zero, andcis the central charge of SVir. [ABSTRACT FROM PUBLISHER]

In this paper, we shall give a refinement of the geometric realization of Lusztig’s algebra f given by Lusztig. This realization gives a geometric interpretation of the decomposition of f. As an application, we shall give geometric realizations of Lusztig’s symmetries on the whole quantum group. [ABSTRACT FROM AUTHOR]

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ℨ-S-semipermutable if H permutes with every Sylow p-subgroup of G in ℨ for all p∉π(H); H is said to be ℨ-S-seminormal if it is normalized by every Sylow p-subgroup of G in ℨ for all p∉π(H). The main aim of this paper is to characterize the ℨ-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-semipermutable in G and the ℨ-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ℨ are ℨ-S-seminormal in G. [ABSTRACT FROM AUTHOR]

In this paper and a sequel, we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz. [ABSTRACT FROM AUTHOR]

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or −1 on C. In this paper, we classify the sign conjugacy classes of alternating groups. [ABSTRACT FROM AUTHOR]

A group G is called mixed identity-free if for every n ∈ ℕ and every w ∈ G ∗ F n , there exists a homomorphism φ : G ∗ F n → G such that φ is the identity on G and φ (w) is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups provided by Ol'shanskii et al. to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite p-groups which are mixed identity-free. [ABSTRACT FROM AUTHOR]

It is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the orderm k-automorphisms ofSL(2,k). We first characterize the forms that orderm k-automorphisms ofSL(2,k) take and then we find simple conditions on matricesAandB, involving eigenvalues and the field that the entries ofAandBlie in, that are equivalent to isomorphy between the orderm k-automorphismsInnAandInnB. We examine the number of isomorphy classes and conclude with examples for selected fields. [ABSTRACT FROM PUBLISHER]

Given a finite groupG, we denote byΔ(G) the commuting graph ofGwhich is defined as follows: the vertex set isGand two distinct verticesxandyare joined by an edge if and only ifxy = yx. Clearly,Δ(G) is always connected for any groupG. We denote byκ(G) the number of spanning trees ofΔ(G). In the present paper, among other results, we first obtain the valueκ(G) for some specific groupsG, such as Frobenius groups, Dihedral groups, AC-groups, etc. Next, we characterize the alternating groupA5, in the class of nonsolvable groups through its tree-numberκ(A5). Finally, we classify the finite groups for which the power graph and the commuting graph coincide. [ABSTRACT FROM PUBLISHER]

The notion of a quantum family of maps has been introduced in the framework ofC -algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept withweak coactionsintroduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction. [ABSTRACT FROM AUTHOR]

This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvableChermak–Delgado simple groups, which the author defines. The notion of “Chermak–Delgado simple,” or “CD-simple” for short, is a generalization of simple groups through the Chermak–Delgado lattice. The author completes a classification of Chermak–Delgado simple groups under certain restrictions on the primes involved in the group order. [ABSTRACT FROM PUBLISHER]

This paper gives an explicit formula for the Hilbert series of algebras of Veronese type. [ABSTRACT FROM AUTHOR]

All groups considered in this paper are finite soluble groups. Let t p ( G ) = max H??G { j|p j ||| H G : H |} and t ( G ) = max p ?p( G ) t p ( G ). In this paper, we study the influence of t ( G ) on the structure of a finite soluble group G . In particular, the bounds of the nilpotent length (derived length, p -length) of a soluble group G with t ( G ) = 2 are found. [ABSTRACT FROM AUTHOR]

In a previous paper, the author, in collaboration with Abhyankar, has proved that under certain assumptions the Galois groups of Carlitz–Drinfeld iterates of linear polynomials are general linear groups. The proof used Cameron–Kantor's characterization of linear groups. In this paper, we recover the result in very generic case by using a more elementary Wagner's characterization of linear groups. This is done using some commutative algebra. [ABSTRACT FROM AUTHOR]

A group is fiat if every finite conjugacy class is a coset of a (necessarily normal) subgroup. This paper is the first to consider the consequences of flatness condition for finite ρ-groups. The flatness condition is weaker than that of Camina pairs (as we shall prove in this paper) and a key aim of this paper is to prove a similar main result to that in the theory of such pairs. Consequences for orders, nilpotence classes and orders of centre of the groups are deduced. [ABSTRACT FROM AUTHOR]

In a recent paper by the authors, the associative and the Lie algebras of Weyl type A[D] = A ⊗ F[D] were introduced, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a field of characteristic 0 and D consisting of locally finite derivations of A, is studied. The isomorphism classes of these associative and Lie algebras are determined. The structure of these algebras is described explicitly. [ABSTRACT FROM AUTHOR]

LetSbe a semigroup and 𝔽 be a field. For an idealJof the semigroup algebra 𝔽[S] ofSover 𝔽, let ϱJdenote the restriction (toS) of the congruence on 𝔽[S] defined by the idealJ. A semigroupSis called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β ofS. In this paper we show that ifSis a semilattice or a rectangular band then φ{S; 𝔽}: J → ϱJis a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (ℬS; ○ ) if and only ifSis a permutable semigroup. [ABSTRACT FROM PUBLISHER]

Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and is a field, then the group ring contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and is a field, then contains no non-trivial units. The support of an element in , denoted by , is the set . In this paper, we study possible zero divisors and units with supports of size 4 in group algebras of torsion-free groups. We prove that if α, β are non-zero elements in for a possible torsion-free group G and an arbitrary field such that and , then . In [J. Group Theory, 16 no. 5, 667-693], it is proved that if is the field with two elements, G is a torsion-free group and such that and , then . We improve the latter result to . Also, concerning the Unit Conjecture, we prove that if for some and , then . [ABSTRACT FROM AUTHOR]

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined. [ABSTRACT FROM PUBLISHER]

Let ℋ = ℋℂ, −1(𝔖n) be the Hecke algebra of the symmetric group 𝔖n. For partitions λ and ν with ν 2 − regular, define the Specht moduleS(λ) and the irreducible moduleD(ν). Definedλν = [S(λ):D(ν)] to be the composition multiplicity ofD(ν) inS(λ). In this paper we compute the decomposition numbersdλνfor all partitions of the form λ = (a,c, 1b) and ν 2 − regular. [ABSTRACT FROM AUTHOR]

The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the Weyl arrangement and their parallel translations. It was introduced by J.-Y. Shi in the study of the Kazhdan–Lusztig representation of the affine Weyl groups. M. Yoshinaga showed that the cone over every Shi arrangement is free. In this paper, we construct an explicit basis for the derivation module of the cone over the Shi arrangements of the typeBℓorCℓ. [ABSTRACT FROM AUTHOR]

In [Tărnăuceanu, M. (2015). Finite groups with a certain number of cyclic subgroups. Amer. Math. Monthly. 122:275–276], Tărnăuceanu described the finite groups G having exactly cyclic subgroups. In [Belshoff, R., Dillstrom, J., Reid, L. Finite groups with a prescribed number of cyclic subgroups. To appear in Communications in Algebra], the authors used elementary methods to completely characterize those finite groups G having exactly cyclic subgroups for Δ = 2, 3, 4 and 5. In this paper, we prove that for any Δ > 0 if G has exactly cyclic subgroups, then and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly cyclic subgroups for. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definition of generalized H-Hom–Lie algebras (i.e., Hom-Lie algebras in the Yetter–Drinfeld category) for any Hopf algebra H, and obtain generalized H-Hom-Lie algebras from H-Hom-associative algebras (i.e., Hom-associative algebras in) and generalizedH-Lie algebras (i.e., Lie algebras in), respectively. We show that if A is a sum of two H-commutative Hom-associative subalgebras, then the H-commutator Hom-ideal of A is nilpotent. Finally, we describe the H-Hom-Lie ideal structures of the H-Hom-associative algebras by analogy with that of H-algebras. [ABSTRACT FROM AUTHOR]

LetKbe an associative and commutative ring with 1,ka subring ofKsuch that 1 ∈ k,Kis an integral finitely generated extension ofk, the element 2 invertible ink, andkis semilocal. The paper studies subgroups of the general linear groupGLn(K) withn ≥ 2 containing the special linear groupSLn(k). [ABSTRACT FROM AUTHOR]

In this paper, we construct derived equivalences between two subrings of relevant Φ-Yoneda rings from an arbitrary short exact sequence in an abelian category. As a consequence, any short exact sequence in an abelian category gives rise to a derived equivalence between two subrings of endomorphism rings. [ABSTRACT FROM PUBLISHER]

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani, and Sochen in [3]. In this paper, closed formulae of the finite split oscillator dictionary 𝔖sare revisited by a simple proof, the structure of nonsplit tori of the groupSL(2, 𝔽p) is studied, and an explicit algorithm for computing the finite nonsplit oscillator dictionary 𝔖nsis described. [ABSTRACT FROM AUTHOR]

LetE/Fbe a Galois extension of number fields with the quaternion Galois groupQ8. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel ofEwith the same orders of some of its subfields. LetE/ℚ be a Galois extension of number fields with the Galois groupQ8andpan odd prime such thatp ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that thep-Sylow subgroup of the tame kernel is nontrivial, thenpr-rank(K2(E/K)) is even, i.e., 2|pr-rank(K2(𝒪E)) − pr-rank(K2(𝒪K)), whereKis the quartic subfield ofE. [ABSTRACT FROM PUBLISHER]

This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers. LetKbe a field, and letAbe an algebra overK. Then the tensor productA ⊗ A = A ⊗KAis also aK-algebra, and it is quite possible that there exists an algebra homomorphism Δ:A → A ⊗ A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence providesAwith a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection ofA-modules, and whenAand Δ satisfy some rather mild axioms, thenAis called a bialgebra. Classical examples of bialgebras include group ringsK[G] and Lie algebra enveloping ringsU(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on HomK(A,A), and this leads quite naturally to the definition of a Hopf algebra. [ABSTRACT FROM AUTHOR]

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed fieldF, whose image contains an almost cyclic matrix of prime-power order. A matrixMis called cyclic if its characteristic and minimum polynomials coincide, and we callMalmost cyclic if, for a suitable α ∈F,Mis similar to diag(α·Idh,M1), whereM1is cyclic and 0 ≤ h ≤ n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements. [ABSTRACT FROM AUTHOR]

In this paper, the structure of locally finite groups of infinite rank whose subgroups of infinite rank are permutable in a subgroup of finite index is investigated. [ABSTRACT FROM AUTHOR]

A group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(픓픉)프) if, and only if, it is non-perfect minimal non-FC, where (픓픉)프 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(픓픉)프) is itself in ((PF)L∪(픓픉)프) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(픓픉)프)-subgroups. [ABSTRACT FROM AUTHOR]

LetHbe a finite group,n≥3, andqbe a prime power such thatq−1 divides neithernnorn−1. Denote byX1(H) the first column of the ordinary character table ofH. LetGbe a finite group such that. In this paper, we will show that if, thenH≅G. As a consequence, we show that the almost simple groupsGare uniquely determined by the structure of their complex group algebras. This extends a result of Tong-Viet [30] to a family of almost simple groups of Lie type of arbitrary rank. [ABSTRACT FROM AUTHOR]

An automorphismαof a groupGis called aweakly power automorphismif it maps every non-periodic subgroup ofGonto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic groupGof derived length at most 3 are power automorphisms, i.e. they fix all subgroups ofG. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism. [ABSTRACT FROM PUBLISHER]

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G12, G24, G25 and G26. Then we give representatives of ail the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups. [ABSTRACT FROM AUTHOR]

This paper is part of a larger programme investigating the B -injectors in arbitrary groups, more precisely, investigating in which groups the B -injectors are conjugate. To answer this question it is critical to have information on the intersection of the B -injectors with the components of the group. Here we are dealing with components having a factor group isomorphic to G 2 ( q ), 3 D 4 ( q ), S z ( q ), or R ( q ). As it is usually quite easy to determine the intersections of B -injectors with components corresponding to Chevally groups of large rank, the groups above among others turn out to be critical in answering the question whether B -injectors are conjugate. [ABSTRACT FROM AUTHOR]

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the influence of permutizers of some subgroups of finite groups on the structure of finite groups. We use permutizers of maximal subgroups as well other simple conditions to characterize supersolubility of finite groups. Also we use permutizers of almost maximal subgroups to get a necessary and sufficient condition for a finite group to be supersoluble, which shows that the permutizer condition Is, in a sense, equivalent to the condition that PG(M)>M if M is an almost maximal subgroup of G. The paper generalizes and deepens the related results in Beidleman and Robinson (1997), Zhang (1986), and Guo (1989). [ABSTRACT FROM AUTHOR]

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If [Xfr] is a class of groups, a group G is said to have [Xfr]-conjugate classes of subgroups if G/CoreG(NG(H)) ∈ [Xfr] for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite). [ABSTRACT FROM AUTHOR]

The groups having exactly one normalizer are well-known. They are the Dedekind groups. All finite groups having exactly two normalizers were classified by M. D. Pérez-Ramos and, in a recent paper, S. Camp-Mora generalized that result to locally finite groups. In this paper, we will characterize arbitrary groups with a finite number of normalizers and we will investigate the properties of arbitrary groups with two, three and four normalizers. [ABSTRACT FROM AUTHOR]

In Everitt [Everitt, B. J. (2000). Alternating quotients of Fuchsian groups. J. Algebra 223: 457–476], it was shown, in particular, that each Fuchsian triangle groupΔ(p, q, r) has among its homomorphic images all but finitely many of the alternating groups. Treating p, q as fixed, the methods of Everitt (2000) give a quadratic function N(r) of r such that An is an image ofΔ(p, q, r) for every integer n ≥ N(r). We conjecture that there is a linear function of r with this property. In this paper, we will show that the conjecture holds for the Fuchsian triangle groupsΔ(3, q, r). [ABSTRACT FROM AUTHOR]