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We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP 1. We prove that the class of c -nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2 < c , the generic c -nilpotent Lie algebra over F p is strictly NSOP 4 and c -dependent. Via the Lazard correspondence, we obtain the same result for c -nilpotent groups of exponent p , for an odd prime p > c. [ABSTRACT FROM AUTHOR]

Let G be a finite solvable group and H a non-normal core-free solvable subgroup of G. We show that if the normalizer of any nontrivial normal subgroup of Fit (H) is equal to H , then H has a nilpotent normal complement K such that G = K H and K Z (Fit (H)) is a Frobenius group. [ABSTRACT FROM AUTHOR]

For finitely generated torsion‐free nilpotent groups, the associated Mal'cev Lie algebra of the group is used frequently when studying the R∞$R_\infty$‐property. Two such groups have isomorphic Mal'cev Lie algebras if and only if they are abstractly commensurable. We show that the R∞$R_\infty$‐property is not invariant under abstract commensurability within the class of finitely generated torsion‐free nilpotent groups by providing counterexamples within a class of 2‐step nilpotent groups associated to edge‐weighted graphs. These groups are abstractly commensurable to 2‐step nilpotent quotients of right‐angled Artin groups. [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]

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G , that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S -permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S -permutable subgroups of G containing A. We prove that if G is a soluble group, then S -permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$ , that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G. [ABSTRACT FROM AUTHOR]

Let 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 Sch U = { G | Sch (G) ⊆ U } , where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class Sch U . In particular, we prove that Sch U is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every Sch U -critical group is a Schmidt group. In addition, we show that Sch U 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 Sch U -critical groups and all minimal non-supersoluble groups that belong to Sch U . [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]

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]

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 a x = 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]

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]

The authors give a brief survey of some results concerning nilpotent braces and their generalizations. Various results concerning *-hypercentral and locally *-nilpotent braces are given. [ABSTRACT FROM AUTHOR]

If θ is a subgroup property, a group G is said to satisfy the double chain condition on θ-subgroups if it admits no infinite double chain · · · < X−n < · · · < X−1 < X0 < X1 < · · · < Xn < · · · consisting of θ-subgroups. Here we want to describe the structure of locally finite and locally nilpotent groups satisfying the double chain condition on non-ascendant subgroups in term of chain conditions and of ascendant subgroups. [ABSTRACT FROM AUTHOR]

Given a finite group G, the coprime graph of G, denoted by Γ(G), is defined as an undirected graph with the vertex set G, and for distinct x,y ∊ G, x is adjacent to y if and only if (o(x),o(y)) = 1, where o(x) and o(y) are the orders of x and y, respectively. This paper classifies the finite groups with split, threshold and chordal coprime graphs, as well as gives a characterization of the finite groups whose coprime graph is a cograph. As some applications, the paper classifies the finite groups G such that Γ(G) is a cograph if G is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a group G is a graph with vertex set G, where two distinct vertices x and y are adjacent if and only if there exists an element w in G such that both x and y are powers of w . To obtain the proper enhanced power graph, we consider the induced subgraph on the set G \ D , where D represents the set of dominating vertices in the enhanced power graph. In this paper, we aim to determine the strong domination number of the proper enhanced power graphs of finite nilpotent groups. [ABSTRACT FROM AUTHOR]

In this paper, we construct odd finitely summable spectral triples based on length functions of bounded doubling on noncommutative solenoids. Our spectral triples induce a Leibniz Lip-norm on the state spaces of the noncommutative solenoids, giving them the structure of Leibniz quantum compact metric spaces. By applying methods of R. Floricel and A. Ghorbanpour, we also show that our odd spectral triples on noncommutative solenoids can be considered as inductive limits of spectral triples on rotation algebras. In the final section, we prove a noncommutative version of Wiener's lemma and show that our odd spectral triples can be defined to have an associated smooth dense subalgebra which is stable under the holomorphic functional calculus, thus answering a question of B. Long and W. Wu. The construction of the smooth subalgebra also extends to the case of nilpotent discrete groups. [ABSTRACT FROM AUTHOR]

We prove that if G and H are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent, then G and H are isomorphic. [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]

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]

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]

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]

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]

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.

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]

In the paper the properties of infinite locally finite groups with non-Dedekind locally nilpotent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic p-subgroups for a unique prime p. In particular, in the paper is prove the following assertions: 1) Let G be an infinite locally finite group and contain the locally nilpotent norm NA G with the non-Hamiltonian Sylow p-subgroup (NAG)p. Then G is a finite extension of a quasicyclic p-subgroup, all Sylow p′-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow q-subgroups (q is an odd prime, q ε π(G), q ≠ p) are cyclic, Sylow 2-subgroups (p ≠ 2) are either cyclic or finite quaternion 2-groups (Theorem 1). 2) Let G be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm NAG of Abelian non-cyclic subgroups. Then G = Gp ≠ H, where Gp is an infinite HAp-group of one of the types (1)-(4) of Proposition 2 in present paper, which coincides with the Sylow p-subgroup of the norm NAG, H is a finite group, all Abelian subgroups of which are cyclic, and (|H|, p) = 1. Any element h λ H of odd order that centralizes some Abelian non-cyclic subgroup M λ NAG is contained in the centralizer of the norm NAG. (Theorem 2). 3) Let G be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm NAG of Abelian non-cyclic subgroups. Then G = H λ K, where H is a finite group, all Abelian subgroups of which are cyclic, (|H|, 2) = 1, K is an infinite 2-group of one of the types (5)-(6) of Proposition 2 (in present paper). Moreover, the norm NAK of Abelian non-cyclic subgroups of the group K is finite, K ∩ NAG = NAK and coincides with the Sylow 2-subgroup (NAG)2 of the norm NA G of a group G. Moreover, any element h ε H of the centralizer of some Abelian non-cyclic subgroup M < NAG is contained in the centralizer of the norm NAG. (Theorem 4). [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]

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]