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Let $R$ be a finitely generated commutative domain and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of a crossed product $R * N$.

A celebrated theorem of Helmut Wielandt shows that the nilpotent residual of the subgroup generated by two subnormal subgroups of a finite group is the subgroup generated by the nilpotent residuals of the subgroups. This result has been extended to saturated formations in Ballester‐Bolinches, Ezquerro, and Pedreza‐Aguilera [Math. Nachr. 239–240 (2002), 5–10]. Although Wielandt's result is not true in arbitrary locally finite groups, we are able to extend it (even in a stronger form) to homomorphic images of periodic linear groups. Also, all results in Ballester‐Bolinches, Ezquerro, and Pedreza‐Aguilera [Math. Nachr. 239–240 (2002), 5–10] are extended to locally finite groups, so it is possible to characterize the class of locally finite groups with a locally nilpotent derived subgroup as the largest subgroup‐closed saturated formation X$\mathfrak {X}$ such that, for all SL$\mathbf {SL}$‐closed saturated formations F$\mathfrak {F}$, the F$\mathfrak {F}$‐residual of an X$\mathfrak {X}$‐group generated by F$\mathfrak {F}$‐subnormal subgroups is the subgroup generated by their F$\mathfrak {F}$‐residuals. Our proofs are based on a reduction theorem that is of an independent interest. Furthermore, we provide strengthened versions of Wielandt's result for other relevant classes of groups, among which we mention the class of paranilpotent groups. A brief discussion on the permutability of the residuals is given at the end of the paper. [ABSTRACT FROM AUTHOR]

In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel)115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 푝-groups, J. Algebra485 (2017), 230–246] using elementary techniques. [ABSTRACT FROM AUTHOR]

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that $G/N'$ is nilpotent, then G itself is nilpotent. We say that a group class is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that $G/N'$ belongs to. Hall classes have been considered by several authors, such as Plotkin ['Some properties of automorphisms of nilpotent groups', Soviet Math. Dokl. 2 (1961), 471–474] and Robinson ['A property of the lower central series of a group', Math. Z. 107 (1968), 225–231]. A further detailed study of Hall classes is performed by us in another paper ['Hall classes of groups', to appear] and we also investigate the behaviour of the class of finite-by- groups for a given Hall class ['Hall classes in linear groups', to appear]. The aim of this paper is to prove that for most natural choices of the Hall class , also the classes $(\mathbf{L}\mathfrak{F})\mathfrak{Y}$ and are Hall classes, where L is the class of locally finite groups and is the class of locally finite groups of finite exponent. [ABSTRACT FROM AUTHOR]

If $G$ is a group, a subgroup $H$ of $G$ is said to be contranormal in $G$ if $H^G = G$, where $H^G$ is the normal closure of $H$ in $G$. We say that a group is contranormal-free if it does not contain proper contranormal subgroups. Obviously, a nilpotent group is contranormal-free. Conversely, if $G$ is a finite contranormal-free group, then $G$ is nilpotent. We study (infinite) groups whose subnormal subgroups are contranormal-free. We prove that if $G$ is a group which contains a normal nilpotent subgroup $A$ such that $G/A$ is a periodic Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is generated by subnormal nilpotent subgroups; in particular $G$ is a Baer group. Furthermore, if $G$ is a group which contains a normal nilpotent subgroup $A$ such that the $0$-rank of $A$ is finite, the set $\Pi(A)$ is finite, $G/A$ is a Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is a Baer group.

This paper addresses Noether’s problem for p-groups G, having an abelian normal subgroup of index p, under the condition Gp={gp:g∈G}≤Z(G)—the center of G. We prove that the fixed field K(G)=K(x(g):g∈G)G is rational over K in such cases, focusing on both the classification and structural analysis of these groups. Our results extend existing work by removing restrictive assumptions and providing a refined understanding of p-groups and their representations.

Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.

For a finite group G, the average order o(G) is defined to be the average of all order elements in G, that is o(G) = ...(x), where o(x) is the order of element x in G. Jaikin-Zapirain in [On the number of conjugacy classes of finite nilpotent groups, Advances in Mathematics, 227 (2011) 1129-1143] asked the following question: if G is a finite (p-) group and N is a normal (abelian) subgroup of G, is it true that o(N)1/2 ≤ o(G)? We say that G satisfies the average condition if o(H) ≤ o(G), for all subgroups H of G. In this paer we show that every finite abelian group satisfies the average condition. This result confirms and improves the question of Jaikin-Zapirain for finite abelian groups. [ABSTRACT FROM AUTHOR]

In the paper we study structure of soluble-by-finite groups of finite torsion-free rank which admit faithful modules with conditions of primitivity. In particular, we prove that under some additional conditions if an infinite finitely generated linear group $G$ of finite rank admits a fully primitive fully faithful module then $G$ has infinite $FC$-centre.

Let N be a nilpotent Lie group and K a compact subgroup of the automorphism group Aut(N) of N. It is well known that if (K ⋉ N , K) is a Gelfand pair then N is at most 2-step nilpotent Lie group. This notion has been generalized to non-compact groups K. In this work, we exhibit a family (K m ⋉ N m , K m) of generalized Gelfand pairs, where N m is a m + 2 -step nilpotent Lie group and K m is isomorphic to R m + 1 . [ABSTRACT FROM AUTHOR]

Let G be a group given by a free presentation G ≅ F / R. The 2-nilpotent multiplier of G is the abelian group (2) (G) = (R ∩ γ 3 (F)) / γ 3 (R , F) which is invariant of G [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc.58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z.226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of (2) (G) , when G is a finite (generalized) extra special p -group. Moreover, the descriptions of the triple tensor product ⊗ 3 G , and the triple exterior product ∧ 3 G are given. [ABSTRACT FROM AUTHOR]

Heineken [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel), 56 no. 5 (1991) 417–423.] studied the order of the nilpotent subgroups of the largest order of a solvable group. We point out an error, and thus refute the proof of the main result of [H. Heineken, Nilpotent subgroups of finite soluble groups, Arch. Math.(Basel), 56 no. 5 (1991) 417–423]. [ABSTRACT FROM AUTHOR]

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is some R 0 > exp(exp(Cd C )) for which the number of elements in a ball of radius R 0 in a Cayley graph of G is bounded by $${R_0^d}$$ , then G has a finite-index subgroup which is nilpotent (of step < C d ). An effective bound on the finite index is provided if “nilpotent” is replaced by “polycyclic”, thus yielding a non-trivial result for finite groups as well.

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group is equal to the growth rate of the induced endomorphism on its abelianization, generalizing the corresponding result for an automorphism in [T. Koberda, Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups, In the Tradition of Ahlfors–Bers. VI, Contemp. Math. 590, American Mathematical Society, Providence 2013, 87–99]. [ABSTRACT FROM AUTHOR]

In 1993, moneyhun showed that if L is a Lie algebra such that dim(L/Z(L)) =n, then dim(L²) ≤ 1/2n(n-1). The author and Saeedi investigated the converse of Moneyhun's result under some conditions .In this paper, We extend their results to obtain several upper bounds for the dimension of a Lie algebra L in terms of dimensions of L², Where L² is the derived subalgebra. Moreover, we give an upper bound for the dimension of the dimension of the c-nilpotent multiplier of a pair of Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for the powers of an interval matrix to be nilpotent. We show an interval matrix A is nilpotent if and only if p(B) = 0, where B is a point matrix, introduced by Mayer (Linear Algebra Appl. 58 (1984) 201-216), constructed by the (*) property. We observed that the spectral radius, determinant, and trace of a nilpotent interval matrix equal zero but in general its converse is not true. Some properties of nonnegative nilpotent interval matrices are derived. We also show that an irreducible interval matrix A is nilpotent if and only if |A| is nilpotent. [ABSTRACT FROM AUTHOR]

Abstract We introduce a special class of powerful p -groups that we call powerfully nilpotent groups that are finite p -groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that leads naturally to a classification in terms of an 'ancestry tree' and powerful coclass. We show that there are finitely many powerfully nilpotent p -groups of each given powerful coclass and develop some general theory for this class of groups. We also determine the growth of powerfully nilpotent groups of exponent p 2 and order p n where p is odd. The number of these is f (n) = p α n 3 + o (n 3) where α = 9 + 4 2 394. For the larger class of all powerful groups of exponent p 2 and order p n , where p is odd, the number is p 2 27 n 3 + o (n 3). Thus here the class of powerfully nilpotent p -groups is large while sparse within the larger class of powerful p -groups. [ABSTRACT FROM AUTHOR]

Abstract Let R be an associative ring with unity and let [ R ] and U (R) denote the associated Lie ring (with [ a , b ] = a b − b a) and the group of units of R , respectively. In 1983 Gupta and Levin proved that if [ R ] is a nilpotent Lie ring of class c then U (R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [ R ] is n -Engel. We construct an algebra R over a field of characteristic ≠ 2 , 3 such that the Lie algebra [ R ] is 5-Engel but the group U (R) is not. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of rings whose elements are a stun of a central element and a nilpotent element, namely, a ring R is called CN if each element a of R has a decomposition a = c + n where c is central and n is nilpotent. In this note, we characterize elements in Mn(R) and U2(R) having CN-decompositions. For any field F. we give examples to show that Mn(F) can not be a CN-ring. For a division ring D. we prove that if A/n(O) is a CN-ring, then the cardinality of the center of D is strictly greater than n. Especially, we investigate several kinds of conditions under which some subrings of full matrix rings over CN rings are CN. [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]

We study the supersolvability of finite groups and the nilpotent length of finite solvable groups under the assumption that all their exactly n-minimal subgroups are S-permutable, where n is an arbitrary integer. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = {σi|i∈I} some partition of the set of all primes. A subgroup A of G is said to be generalized σ-subnormal in G if A = ⟨L,T⟩, where L is a modular subgroup and T is a σ-subnormal subgroup of G. In this paper, we prove that if every Schmidt subgroup of G is generalized σ-subnormal in G, then the commutator subgroup G′ of G is σ-nilpotent. [ABSTRACT FROM AUTHOR]

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a1,..., an) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3. [ABSTRACT FROM AUTHOR]

In this paper we study the homogeneity of radicals defined by nilpotence or primality conditions, in rings graded by a semigroup S . When S is a unique product semigroup, we show that the right (and left) strongly prime and uniformly strongly prime radicals are homogeneous, and an even stronger result holds for the generalized nilradical. We further prove that rings graded by torsion-free, nilpotent groups have homogeneous upper nilradical. We conclude by showing that non-semiprime rings graded by a large class of semigroups must always contain nonzero homogeneous nilpotent ideals. [ABSTRACT FROM AUTHOR]

Let x be an element of a group G . For a positive integer n let E n ( x ) be the subgroup generated by all commutators [ . . . [ [ y , x ] , x ] , … , x ] over y ∈ G , where x is repeated n times. There are several recent results showing that certain properties of groups with small subgroups E n ( x ) are close to those of Engel groups. The present article deals with orderable groups in which, for some n ≥ 1 , the subgroups E n ( x ) are polycyclic. Let h ≥ 0 , n > 0 be integers and G an orderable group in which E n ( x ) is polycyclic with Hirsch length at most h for every x ∈ G . It is proved that there are ( h , n ) -bounded numbers h ⁎ and c ⁎ such that G has a finitely generated normal nilpotent subgroup N with h ( N ) ≤ h ⁎ and G / N nilpotent of class at most c ⁎ . [ABSTRACT FROM AUTHOR]

Let q be a prime power and F q n be the finite field with q n elements, where n > 1 . We introduce the class of the linearized polynomials L ( X ) over F q n such that L ( t ) ( X ) : = L ∘ L ∘ ⋯ ∘ L ︸ t times ( X ) ≡ 0 ( mod X q n − X ) for some t ≥ 2 , called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to obtain permutations of F q n from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle decomposition. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers. [ABSTRACT FROM AUTHOR]

LetAbe a nilpotent matrix of index three and consider the Yang–Baxter-like matrix equationAXA=XAX. We first obtain a simplified matrix equation of the same type withAreplaced by its Jordan form, whose Jordan blocks are at most. Then we obtain a system of matrix equations of smaller sizes to find all the commuting solutions of the original matrix equation. [ABSTRACT FROM PUBLISHER]

H is called an ℳ-embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H ∈ Syl( B) and B is ℳ-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ℳ-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ⩽ | P| and d divides | P|. If every subgroup H of P with | H| = d is ℳ-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/ C is cyclic of order 5, (2) I/C is 5′-group, (3) I/ C ≅ A . [ABSTRACT FROM AUTHOR]

We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-0 coefficients.We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for SL(n) with coefficients of arbitrary characteristic and for SO(n) and Sp(2n) with characteristic-2 coefficients. [ABSTRACT FROM AUTHOR]

The non commuting graph of a non-abelian finite group $G$ is defined as follows: its vertex set is $G-Z(G)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. In this paper we prove some new results about this graph. In particular we will give a new proof of theorem 3.24 of [2]. We also prove that if $G_1$, $G_2$, ..., $G_n$ are finite groups such that $Z(G_i)=1$ for $i=1,2,...,n$ and they are characterizable by non commuting graph, then $G_1times ...times G_n$ is characterizable by non commuting graph.

Let H be a nilpotent subgroup of ΓL_n (q) = GL_n (q), where φ denotes the field automorfism induced by the Frobenius map. We give a condition on the primes dividing |H ∩ GL_n (q)| under which H is conjugate to a subgroup of the generalized monomial group Diag_n (F∗_q ) Sym(n).We show an application of this result to the determination of Carter subgroups of finite groups.

We give a combinatorial description of the finitely generated free weak nilpotent minimum algebras and provide explicit constructions of normal forms. [ABSTRACT FROM AUTHOR]

Given a finite group G with socle isomorphic to L ( q), q ≥ 4, we describe, up to conjugacy, all pairs of nilpotent subgroups A and B of G such that A ∩ B ≠ 1 for all g ∈ G. [ABSTRACT FROM AUTHOR]

We expose fundamentals of the theory of varieties of nilpotent MR-groups and compare various definitions of nilpotency in this category. [ABSTRACT FROM AUTHOR]

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A which depends only on the Grothendieck ring of H. When H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are. [ABSTRACT FROM AUTHOR]

Let $${ \mathcal {F}}$$ be a saturated formation and G a finite group such that $${N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}$$ for every subgroup H of G. If the minimal non- $${ \mathcal {F}}$$ -group is soluble, then $${G \in \mathcal {F}}$$ . [ABSTRACT FROM AUTHOR]

An SIR type of compartmental model with a standard incidence rate and a nonlinear recovery rate was formulated to study the impact of available resources of public health system especially the number of hospital beds. Cusp, focus and elliptic type of nilpotent singularities of codimension 3 are discovered and analyzed in this three dimensional model. Complex dynamics of disease transmission including multi-steady states and multi-periodicity are revealed by bifurcation analysis. Large-amplitude oscillations found in our model provide a more reasonable explanation for disease recurrence. With clinical data, our studies have practical implications for the prevention and control of infectious diseases. [ABSTRACT FROM AUTHOR]

In an earlier paper, the authors proved that a process described much earlier for passing from a finitely generated nilpotent group N of a certain kind to a nilpotent space X of finite type produced a bijection of Mislin genera 𝒢(N)≅𝒢(X). The present paper is concerned with related results obtained by weakening the restrictions on N and generalizing the homotopical nature of the spaces X to be associated with a given N.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a finite subset of an ordered group that generates a nonabelian ordered group, then $|S^2|\geq 3|S|-2$. This generalizes a classical result from the theory of set addition. [ABSTRACT FROM PUBLISHER]

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to formulate and prove an L- L analog of Miyachi's theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy's and Cowling-Price's uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where G is further assumed to be simply connected, p = 2, and q = +∞. When G is more generally exponential solvable, such a principle also holds provided that the center of G is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs. [ABSTRACT FROM AUTHOR]