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The directed power graph P → (G) of a group G is the simple digraph with vertex set G such that x → y if y is a power of x. The power graph of G , denoted by P (G) , is the underlying simple graph. The enhanced power graph P e (G) of G is the simple graph with vertex set G in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any n ≥ 0 and prime numbers p and q , every group of order p n q and p 2 q 2 has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs. Finally, we briefly paid attention to the difference graph, which is a difference between the enhanced graph and power graph and proved that groups whose difference graphs are perfect, have perfect enhanced graphs. [ABSTRACT FROM AUTHOR]

In the framework of weak congruence lattices, many classes of groups have been characterized up to now, in completely lattice-theoretic terms. In this note, the center of the group is captured lattice-theoretically and nilpotent groups are characterized by lattice properties. [ABSTRACT FROM AUTHOR]

Let G be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual p -finiteness of G for proving that this generalized free product is residually a finite nilpotent group or residually a finite metanilpotent group. This approach can be applied under most of the conditions on the amalgamated subgroup that allow the study of residual p -finiteness. Namely, we consider the cases where the amalgamated subgroup is a) periodic, b) locally cyclic, c) central in one of the free factors, d) normal in both free factors, or e) is a retract of one of the free factors. In each of these cases, we give certain necessary and sufficient conditions for G to be residually a) a finite nilpotent group, b) a finite metanilpotent group. [ABSTRACT FROM AUTHOR]

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra \mathbb {Q}G for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in \mathbb {Q}G, we describe a subgroup of finite index in the group of units \mathcal {U}(\mathbb {Z}G) of the integral group ring \mathbb {Z}G that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable. [ABSTRACT FROM AUTHOR]

A ring R is said to be centrally essential if either R is commutative or for any non-central element a ∈ R, there exist two non-zero central elements x and y with ax = y. In this paper, we study centrally essential rings which are nilpotent as Lie rings (centrally essential rings are not necessarily nilpotent as Lie rings, since it is known that they are not necessarily PI rings). It is proved that the following rings are Lie nilpotent: right Artinian centrally essential rings with finitely generated adjoint group, semiperfect centrally essential rings with torsion group of units and nilpotent Jacobson radical, the right uniserial centrally essential rings. [ABSTRACT FROM AUTHOR]

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]

We calculate the conjugacy growth of 2-step free nilpotent groups, extending a previous analysis of Guba and Sapir of the conjugacy growth of the Heisenberg group. [ABSTRACT FROM AUTHOR]

Let be a group, let be an isomorphism of onto a subgroup of , and let be a descending HNN-extension of corresponding to . The potency of is not inherited by even in the simplest case, when is an infinite cyclic group. We prove that if is a finitely generated torsion-free nilpotent group (a polycyclic group); then the index of in is finite and is -potent (virtually -potent), where is the set of all primes greater than . We also prove some generalizations of this assertion. Some of the results of this work on the potency of descending HNN-extensions are analogs of the well-known theorems on the residual finiteness of the HNN-extensions. [ABSTRACT FROM AUTHOR]

For a nilpotent group G without π -torsion, and x , y ∈ G , if x n = y n for a π -number n , then x = y ; if x m y n = y n x m for π -numbers m , n , then x y = y x. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let m be a given positive integer and A a complex square matrix satisfying that (i) all eigenvalues of A are nonnegative, and (ii) rank ⁡ A 2 = rank ⁡ A ; then A has a unique m -th root X with rank ⁡ X 2 = rank ⁡ X , all eigenvalues of X are nonnegative, and moreover there is a polynomial f (λ) with X = f (A). In addition, let A and B be complex n × n matrices with all eigenvalues nonnegative, and rank ⁡ A 2 = rank ⁡ A , rank ⁡ B 2 = rank ⁡ B ; then (i) A = B when A r = B r for some positive integer r , and (ii) A B = B A when A s B t = B t A s for two positive integers s and t. [ABSTRACT FROM AUTHOR]

We use a constructive method to obtain all but finitely many 푝-local representation zeta functions of a family of finitely generated nilpotent groups M n with maximal nilpotency class. For representation dimensions coprime to all primes p < n , we construct all irreducible representations of M n by defining a standard form for the matrices of these representations. [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]

In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed. [ABSTRACT FROM AUTHOR]

Let ΓG denote a graph associated with a group G. A compelling question about finite groups asks whether or not a finite group H must be nilpotent provided ΓH is isomorphic to ΓG for a finite nilpotent group G. In the present work we analyze the problem for different graphs that one can associate with a finite group, both reporting on existing answers and contributing to new ones. [ABSTRACT FROM AUTHOR]

For a group G and a positive integer n write B n (G) = { x ∈ G : | x G | ≤ n }. If s ≥ 1 and w is a group word, say that G satisfies the (n , s) -covering condition with respect to the word w if there exists a subset S ⊆ G such that | S | ≤ s and all w -values of G are contained in B n (G) S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results. Let w be a multilinear commutator word on k variables and let G be a group satisfying the (n , s) -covering condition with respect to the word w. Then G has a soluble subgroup T such that [ G : T ] and the derived length of T are both (k , n , s) -bounded. (Theorem 1.1.) Let k ≥ 1 and G be a group satisfying the (n , s) -covering condition with respect to the word γ k . Then (1) γ 2 k − 1 (G) has a subgroup T such that [ γ 2 k − 1 (G) : T ] and | T ′ | are both (k , n , s) -bounded; and (2) G has a nilpotent subgroup U such that [ G : U ] and the nilpotency class of U are both (k , n , s) -bounded. (Theorem 1.2.) [ABSTRACT FROM AUTHOR]

Abstract–In this paper, based on the classifications of Kn-simple groups for n≤6 and under the case that every maximal subgroup has a π-nilpotent trace, we obtained the necessary and sufficient conditions on solvable groups and proved that two unsolvable problems involving groups, rings and fields are true. [ABSTRACT FROM AUTHOR]

We present an algorithm determining the existence of an isomorphic embedding of one finitely generated partially commutative nilpotent group of class in another. It is shown how such embeddings can be constructed. We also describe an algorithm determining the existence of an embedding of a finitely generated partially commutative nilpotent group of arbitrary class with respect to a graph of nonzero radius in a free nilpotent group of class . [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]

The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in (Inf. Comput. 178(1), 253–262, 2002) where they showed that this problem is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups G having a nilpotent normal subgroup H with nilpotent factor G/H. This paper shows that such a normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails. [ABSTRACT FROM AUTHOR]

We show that every compact aspherical Sasakian manifold with nilpotent fundamental group is diffeomorphic to a Heisenberg nilmanifold. [ABSTRACT FROM AUTHOR]

Assume that H is a finite group that has a normal 2 -complement. Under some conditions, it is proven that the normalizer property holds for H. In particular, if there is a nilpotent subgroup of index 2 in H , then H has the normalizer property. The result of Li, Sehgal and Parmenter, stating that the normalizer property holds for finite groups that have an abelian subgroup of index 2 is generalized. [ABSTRACT FROM AUTHOR]

For each left-invariant semi-Riemannian metric g on a Lie group G, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of g. When the adjoint representation of G satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any g. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products K \ltimes _\rho \mathbb {R}^n, where K is the direct product of a compact and an abelian Lie group and \rho (K) is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the absence of linear growth and suggest new questions. [ABSTRACT FROM AUTHOR]

A group G is called logically generated by a subset S, if every element of G can be defined by a first order formula with parameters from S. We consider the case where G is a direct product of finite nilpotent groups with mutually coprime orders and we show that logical and algebraic generations are equivalent in G. We also prove that in the case when G is a free non-abelian group, if S logically generates G then either it generates G algebraically or 〈 S 〉 is not a free factor of G. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to compute the probability Pr(G) that two elements of the group G, drawn at random with replacement, commute; that is, Pr(G) = Number of ordered pairs (x, y) ∈ G × G such that xy = yx/|G × G| = |G|² In particular, we compute Pr(G) for some groups such as the extraspecial groups of order p³, p prime, for the permutation groups G = Sn and G = An, n ≥ 5, for 10 non-abelian groups of order p4 and for simple groups of certain type. [ABSTRACT FROM AUTHOR]

Let 픛 be a class of finite groups. A group G is called a minimal non-픛-group or simply an 픛-critical group, if G is not in 픛 but all proper subgroups of G are in 픛. An 픑-critical group, for the class 픛 = 픑 all finite nilpotent groups, is called a Schmidt group. We say that a formation 픉 has the Shemetkov property if every 픉-critical group is either a Schmidt group or a cyclic group of prime order. In this paper, properties of the lattice of all soluble subgroup-closed local formations with Shemetkov property are investigated. [ABSTRACT FROM AUTHOR]

Let G be a finite nilpotent group. We denote by γ(G) the number of conjugacy classes of noncyclic subgroups of G, Con(G) the number of conjugacy classes of subgroups of G. In this paper, we give the formula for γ(G), and prove that Con(G) − γ(G) ≥ 4 when the order of G is prime power. Moreover, we get the minimum of γ(G) according to t, where t is the number of distinct noncyclic Sylow subgroups of G. [ABSTRACT FROM AUTHOR]

Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian 2-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively. [ABSTRACT FROM AUTHOR]

We consider descending HNN extensions 〈B,t : t−1Ht = K〉, where B = H is abelian. We show that class-preserving automorphisms of descending HNN extensions are all inner if B is abelian and K≠B. It follows that outer automorphism groups of such conjugacy separable descending HNN extensions are residually finite. [ABSTRACT FROM AUTHOR]

The power word problem for a group G asks whether an expression u 1 x 1 ⋯ u n x n , where the u i are words over a finite set of generators of G and the x i binary encoded integers, is equal to the identity of G . It is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over G ). We start by showing some easy results concerning the power word problem. In particular, the power word problem for a group G is uNC 1 -many-one reducible to the power word problem for a finite-index subgroup of G . For our main result, we consider graph products of groups that do not have elements of order two. We show that the power word problem in a fixed such graph product is AC 0 -Turing-reducible to the word problem for the free group F 2 and the power word problems of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups C without order two elements, the uniform power word problem in a graph product can be solved in AC 0 [ C = L UPowWP (C) ] , where UPowWP (C) denotes the uniform power word problem for groups from the class C . As a consequence of our results, the uniform knapsack problem in right-angled Artin groups is NP -complete. The present paper is a combination of the two conference papers (Lohrey and Weiß 2019b, Stober and Weiß 2022a). In Stober and Weiß (2022a) our results on graph products were wrongly stated without the additional assumption that the base groups do not have elements of order two. In the present work we correct this mistake. While we strongly conjecture that the result as stated in Stober and Weiß (2022a) is true, our proof relies on this additional assumption. [ABSTRACT FROM AUTHOR]

This article, titled "A Finite Group with a Maximal Miller–Moreno Subgroup," discusses the properties and structure of Miller–Moreno groups, which are finite non-Abelian groups where all proper subgroups are Abelian. The article presents a theorem that explores the solvability of these groups based on the properties of their maximal subgroups. It also discusses the concept of Sylow towers and provides lemmas that are used in the proof of the theorem. The article concludes with a remark on the sharpness of the derived length estimate in the theorem. The research was funded by the Belarusian Republican Foundation for Basic Research. [Extracted from the article]

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group G be presented as the quotient of a free group F by a normal subgroup R. Given a positive integer c, the c-nilpotent multiplier of the group G is the abelian group M (c) (G) = (R ∩ γ c + 1 (F)) / γ c + 1 (R , F) . In particular, M (1) (G) is the Schur multiplier of G. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups. [ABSTRACT FROM AUTHOR]

It is well known that if G is a group and H is a normal subgroup of G of finite index k, then x k ∈ H for every x ∈ G . We examine finite groups G with the property that x k ∈ H for every subgroup H of G, where k is the index of H in G. We prove that a finite group G satisfies this property if and only if G is nilpotent. [ABSTRACT FROM AUTHOR]

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an O (1) -generated solvable group with solvability class poly log log n. This notably includes instances where the complement is an O (1) -generated nilpotent group. This problem was previously known to be in P [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to L [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent p > 2 [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N.  Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of q = 1. The general theme is that some counting appears necessary to place G r o u p I s o m o r p h i s m into P. [ABSTRACT FROM AUTHOR]

We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete automorphism group of the direct product. [ABSTRACT FROM AUTHOR]

Assume G is a nilpotent group of class > 3 in which every proper subgroup has class at most 3. In this note, we give the exact upper bound of class of G. [ABSTRACT FROM AUTHOR]

Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

Let G be a non-abelian group and Z (G) be the center of G. The non-commuting graph Γ G associated to G is the graph whose vertex set is G ∖ Z (G) and two distinct elements x , y are adjacent if and only if x y ≠ y x. We prove that if G and H are non-abelian groups with isomorphic non-commuting graphs, such that G is nilpotent, then H is nilpotent, provided | Z (G) | ≥ | Z (H) |. Actually we prove conjecture 3 proposed in V. Grazian and C. Monetta (2023) [5]. [ABSTRACT FROM AUTHOR]

It is shown that the commuting probability of a finite ring cannot be a fraction with square-free denominator, resolving a conjecture of Buckley and MacHale. [ABSTRACT FROM AUTHOR]

A subset D of a group G is a determining set of G if every automorphism of G is uniquely determined by its action on D , and the determining number of G , α (G) , is the cardinality of a smallest determining set. A group G is called a DEG-group if α (G) equals γ (G) , the generating number of G. Our main results are as follows. Finite groups with determining number 0 or 1 are classified; finite simple groups and finite nilpotent groups are proved to be DEG-groups; for a given finite group H , there is a DEG-group G such that H is isomorphic to a normal subgroup of G and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups; for any integer k ≥ 2 , there exists a group G such that α (G) = 2 and γ (G) ≥ k. [ABSTRACT FROM AUTHOR]

One of the most critical concepts in the study of groups is the notion of nilpotency. Nilpotent groups arise in Galois theory, as well as in the classification of groups. In this paper, we apply the notion of nilpotency on B C K -algebras by using fuzzy ideals. First we define the concept of nilpotent fuzzy ideal in B C K -algebras and investigate their properties. Furthermore, we find some relation between fuzzy nilpotent ideals and quotient B C K -algebras. Specially, we construct a nilpotent B C K -algebra by using a fuzzy nilpotent ideal. Finally, we prove that any minimal ideals of a B C K -algebra X , is contained in the center of any fuzzy nilpotent ideals of X. [ABSTRACT FROM AUTHOR]

In this paper, we consider essential idempotents in the finite semisimple group algebra of a nilpotent group, studying conditions for their existence and other implications. Also, we discuss conditions for nilpotent group codes to be equivalent to Abelian ones. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x,y are adjacent if x,y ∈〈z〉 for some z ∈ G. An L(2, 1)-labeling of graph Γ is an integer labeling of V (Γ) such that adjacent vertices have labels that differ by at least 2 and vertices distance 2 apart have labels that differ by at least 1. The λ-number of Γ, denoted by λ(Γ), is the minimum range over all L(2, 1)-labelings. In this paper, we study the lambda number of the enhanced power graph 풫E(G) of the group G. This paper extends the corresponding results, obtained in [X. Ma, M. Feng and K. Wang, Lambda number of the power graph of a finite group, J. Algebraic Combin. 53(3) (2021) 743–754], of the lambda number of power graphs to enhanced power graphs. Moreover, for a nontrivial simple group G of order n, we prove that λ(풫E(G)) = n if and only if G is not a cyclic group of order n ≥ 3. Finally, we determine the lambda number of the enhanced power graphs of nilpotent groups. [ABSTRACT FROM AUTHOR]

From work of Vaughan-Lee in [12] it follows that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan-Lee gives an explicit example of a nilpotent loop of order 12 that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time. [ABSTRACT FROM AUTHOR]

A group G is said to be a T-group if normality in G is a transitive relation. Clearly, as a simple group has the property T, it follows that T is not subgroup closed. A group G is called a T-group if all its subgroups are T-groups. In this note the structure of groups all of whose (proper) subgroups either are nilpotent or satisfy the property T will be investigated. [ABSTRACT FROM AUTHOR]

We propose a method for solving the hidden subgroup problem in nilpotent groups. The main idea is iteratively transforming the hidden subgroup to its images in the quotient groups by the members of a central series, eventually to its image in the commutative quotient of the original group, and then using an abelian hidden subgroup algorithm to determine this image. Knowing this image allows one to descend to a proper subgroup unless the hidden subgroup is the full group. The transformation relies on finding zero sum subsequences of sufficiently large sequences of vectors over finite prime fields. We present a new deterministic polynomial time algorithm for the latter problem in the case when the size of the field is constant. The consequence is a polynomial time exact quantum algorithm for the hidden subgroup problem in nilpotent groups having constant nilpotency class and whose order only have prime factors also bounded by a constant. [ABSTRACT FROM AUTHOR]

Let G be a finite group with the identity element e. The proper order graph of G, denoted by S 풞. (G), is an undirected graph with a vertex set G\ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of 풞. *(G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups. [ABSTRACT FROM AUTHOR]

Let 퐺 and 퐻 be residually nilpotent groups. Then 퐺 and 퐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 퐻 is para-퐺 if there exists a monomorphism of 퐺 into 퐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually nilpotent groups and find sufficient conditions on the monomorphism so that 퐻 is para-퐺. We then prove that, for certain polycyclic groups, if 퐻 is para-퐺, then 퐺 and 퐻 have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic. [ABSTRACT FROM AUTHOR]

Based on some known researches about some self-centralizing subgroups of finite groups being a TI-subgroup or subnormal, we indicate that the following two conditions hypotheses are equivalent: Suppose that G is a finite group, then every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of G is a TI-subgroup or subnormal if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of G is a TI-subgroup or subnormal, which implies that the self-centralizing hypothesis can be removed from some known results. [ABSTRACT FROM AUTHOR]

Let G be a finite permutation group on Ω. An ordered sequence (ω 1 , ... , ω t) of elements of Ω is an irredundant base for G if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same cardinality, G is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order. [ABSTRACT FROM AUTHOR]

An algorithm is presented that determines the maximum rank of a free nilpotent metabelian or, respectively, nilpotent group isomorphically embeddable into a given partially commutative nilpotent group of the same degree of nilpotency. It is shown how these embeddings are realized. [ABSTRACT FROM AUTHOR]

Hom–Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom–Lie algebras are studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well-developed. We establish a theory of the solvable and nilpotent Hom–Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research. Dedicated to Al Farouk School & Kinder garten-Irbid-Jordan [ABSTRACT FROM AUTHOR]