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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]

For a finite group G, let s(G) denote the number of subgroups of G and d(G) denote the number of divisors of |G|. In this paper, we prove that there does not exist a finite group G such that d (G) < s (G) < d (G) + p 1 , where p 1 is the smallest prime divisor of |G|, and determine the structure of finite groups G such that s (G) = d (G) + p 1 . [ABSTRACT FROM AUTHOR]

The co-maximal subgroup graph Γ (G) of a group G is a graph whose vertices are non-trivial proper subgroups of G and two vertices H and K are adjacent if H K = G . In this paper, we continue the study of Γ (G) , especially when Γ (G) has isolated vertices. We define a new graph Γ ∗ (G) , which is obtained by removing isolated vertices from Γ (G) . We characterize when Γ ∗ (G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G. [ABSTRACT FROM AUTHOR]

We consider groups G such that the set [ G , φ ] = { g - 1 g φ | g ∈ G } is a subgroup for every automorphism φ of G, and we prove that there exists such a group G that is finite and nilpotent of class n for every n ∈ N . Then there exists an infinite not nilpotent group with the above property and the Conjecture 18.14 of Khukhro and Mazurov (The Kourovka Notebook No. 20, 2022) is false. [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 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]

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]

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]

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]

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]

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 퐺 be a group and 푘 a positive integer. We say that 퐺 is conjugate separable abelian (CSA) if every maximal abelian subgroup of 퐺 is malnormal. In this paper, as a natural generalization, we study groups with the property that all maximal nilpotent subgroups of class at most 푘 are malnormal, which we refer to as CSN푘 groups, and we show that they have many properties in common with the more widely studied CSA groups. In addition, we introduce the class of nilpotency transitive groups of class 푘, denoted NT푘, and in the presence of a special residuality condition, we prove that the CSN푘 and NT푘 properties are equivalent. [ABSTRACT FROM AUTHOR]

AbstractLet F be a class of finite groups. A group G is called a minimal non-F-group or simply an F-critical group, if G is not in F but all proper subgroups of G are in F. Let Sch(G) denotes the set of all Schmidt subgroups of a group G (i.e. N-critical subgroups of G, where N is the class of all nilpotent groups), and SchU={G|Sch(G)⊆U}, where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class SchU. In particular, we prove that SchU is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every SchU-critical group is a Schmidt group. In addition, we show that SchU is locally defined by the formation function f such that f(p)=S{p}∪π(p−1) for every prime p. Besides, we describe all SchU-critical groups and all minimal non-supersoluble groups that belong to SchU. [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]

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold. [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]

Let KG be the modular group algebra of a group G over a field K of characteristic p > 0. Recently, we have seen the classification of group algebras KG with upper Lie nilpotency index tL(KG) up to 10p - 8. In this paper, our aim is to classify the modular group algebra KG with upper Lie nilpotency index 11p - 9, for G'= γ2(G) as an abelian group. [ABSTRACT FROM AUTHOR]

Let G be a finite group and s (G) denote the number of subgroups of G. Aivazidis and Müller proved that if G is a non-cyclic p-group of order p λ , then s (G) ≥ 6 whenever p λ = 2 3 ; s (G) ≥ (p + 1) (λ - 1) + 2 whenever p λ ≠ 2 3 . In this paper, we generalize the results of Aivazidis and Müller on all finite non-cyclic nilpotent groups. Lower bounds on s (G) of non-cyclic nilpotent groups G are established. [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]

In this paper we prove that if G is a finite group, then the k-th term of the lower central series is nilpotent if and only if for every γk-values x, y ∈ G with coprime orders, either π(o(x)o(y)) ⊆ π(o)x y)) or o(x)o(y)) ≤ o(x y). We obtain an analogous version for the derived series of finite solvable groups, but replacing γk-values by δk-values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements. [ABSTRACT FROM AUTHOR]

Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms g,h:Σ∗→Δ∗$g, h\colon \Sigma ^*\rightarrow \Delta ^*$ there exists any non‐trivial x∈Σ∗$x\in \Sigma ^*$ such that g(x)=h(x)$g(x)=h(x)$. PCP for a group Γ$\Gamma$ takes pairs of group homomorphisms g,h:F(Σ)→Γ$g, h\colon F(\Sigma)\rightarrow \Gamma$ instead, and similarly asks whether there exists an x$x$ such that g(x)=h(x)$g(x)=h(x)$ holds for non‐elementary reasons. The restrictions imposed on x$x$ in order to get non‐elementary solutions lead to several interpretations of the problem; we mainly consider the natural restriction asking that x∉ker(g)∩ker(h)$x \notin \ker (g) \cap \ker (h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic Γ$\Gamma$, but decidable when Γ$\Gamma$ is virtually nilpotent, en route also studying this problem for finite extensions. We also consider a different interpretation of the PCP due to Myasnikov, Nikolaev and Ushakov, proving decidability for torsion‐free nilpotent groups. [ABSTRACT FROM AUTHOR]