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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]

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 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]

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]

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]

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]

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]

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]

In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation .M;F /. If M is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If M has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well. [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]

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]

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]

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]

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]

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: F n ⋊ φ ℤ. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for n = 2 the length of the lower central series of F n ⋊ φ ℤ is equal to 2, ω or ω 2 . [ABSTRACT FROM AUTHOR]

We prove that asymptotic cones of Helly graphs are countably hyperconvex. We use this to show that virtually nilpotent Helly groups are virtually abelian and to characterize virtually abelian Helly groups via their point groups. In fact, we do this for the more general class of coarsely injective spaces and groups. We apply this to prove that the 3‐3‐3‐Coxeter group is not Helly (nor even coarsely injective), thus obtaining the first example of a systolic group that is not Helly, answering a question of Chalopin, Chepoi, Genevois, Hirai, and Osajda. [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]

Given a finite group \mathrm {G}, the faithful dimension of \mathrm {G} over \mathbb {C}, denoted by m_\mathrm {faithful}(\mathrm {G}), is the smallest integer n such that \mathrm {G} can be embedded in \mathrm {GL}_n(\mathbb {C}). Continuing the work initiated by Bardestani et al. [Compos. Math. 155 (2019), pp. 1618–1654], we address the problem of determining the faithful dimension of a finite p-group of the form \mathscr {G}_R≔\exp (\mathfrak {g}_R) associated to \mathfrak {g}_R≔\mathfrak {g}\otimes _\mathbb {Z}R in the Lazard correspondence, where \mathfrak {g} is a nilpotent \mathbb {Z}-Lie algebra and R ranges over finite truncated valuation rings. Our first main result is that if R is a finite field with p^f elements and p is sufficiently large, then m_\mathrm {faithful}(\mathscr {G}_R)=fg(p^f) where g(T) belongs to a finite list of polynomials g_1,\ldots,g_k, with non-negative integer coefficients. The latter list of polynomials is uniquely determined by the Lie algebra \mathfrak {g}. Furthermore, for each 1\le i\leq k the set of pairs (p,f) for which g=g_i is a finite union of Cartesian products \mathscr P\times \mathscr F, where \mathscr P is a Frobenius set of prime numbers and \mathscr F is a subset of \mathbb N that belongs to the Boolean algebra generated by arithmetic progressions. Previously, existence of such a polynomial-type formula for m_\mathrm {faithful}(\mathscr {G}_R) was only established under the assumption that either f=1 or p is fixed. Next we formulate a conjectural polynomiality property for the value of m_\mathrm {faithful}(\mathscr {G}_R) in the more general setting where R is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras \mathfrak {g} that are defined by partial orders, m_\mathrm {faithful}(\mathscr {G}_R) is given by a single polynomial-type formula. Finally, we compute m_\mathrm {faithful}(\mathscr {G}_R) precisely in the case where \mathfrak {g} is the free metabelian nilpotent Lie algebra of class c on n generators and R is a finite truncated valuation ring. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H G of H in G. In this paper, we investigate the structure of a finite group G under the assumption that certain 2-minimal subgroups and 2-maximal subgroups are Hall normally embedded in G, respectively. Some conditions for a finite group to be p-nilpotent and supersolvable are given. [ABSTRACT FROM AUTHOR]

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang–Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces. [ABSTRACT FROM AUTHOR]

Abstract–We prove a few results about non-nilpotent graphs of symmetric groups Sn – namely that they satisfy a conjecture of Nongsiang and Saikia (which is likewise proved for alternating groups An), and that for n ⩾ 19 each vertex has degree at least n ! 2 . We also show that the class of non-nilpotent graphs does not have any "local" properties, i.e. for every simple graph X there is a group G, such that its non-nilpotent graph contains X as an induced subgraph. [ABSTRACT FROM AUTHOR]

Let T denote the class of finitely generated torsion-free nilpotent groups. For a group G let F (G) be the set of isomorphism classes of finite quotients of G. Pickel proved that if G ∈ T , then the set g (G) of isomorphism classes of groups H ∈ T with F (G) = F (H) is finite. We give an explicit description of the sets g (G) for the T -groups G of Hirsch length at most 5. Based on this, we show that for each Hirsch length n ≥ 4 and for each m ∈ N there is a T -group G of Hirsch length n with | g (G) | ≥ m. [ABSTRACT FROM AUTHOR]

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if G and H are finite groups with isomorphic non-commuting graphs and G is nilpotent, then H must be nilpotent as well (Conjecture 2). We characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 for finite AC-groups whenever | Z (G) | ≥ | Z (H) |. [ABSTRACT FROM AUTHOR]

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras. [ABSTRACT FROM AUTHOR]

The authors prove the following results: Let G be a locally graded group and suppose that every non-nilpotent subgroup of G is permutable. If G is torsion-free, then G is nilpotent. If G is non-periodic, then G is soluble. [ABSTRACT FROM AUTHOR]

In [Finite solvable groups with exactly two nonlinear non-faithful irreducible characters, J. Algebra Appl. 18(5) (2019) 1950091], the first author studied directly indecomposable, solvable groups with exactly two nonlinear non-faithful irreducible characters. In this paper, we study the decomposable case and give a classification of decomposable groups possessing exactly two nonlinear non-faithful irreducible characters. In particular, nilpotent groups with this property are also classified. [ABSTRACT FROM AUTHOR]

A rather natural way for trying to obtain a lattice-theoretic characterization of a class of groups ω is to replace the concepts appearing in the definition of ω by lattice-theoretic concepts. The first to use this idea were Kontorovič and Plotkin who in 1954 introduced the notion of modular chain in a lattice, as translation of a central series of a group, to determine a lattice-theoretic characterization of the class of torsion-free nilpotent groups. The aim of this paper is to present a recent application of this translation method to some generalized nilpotency properties. [ABSTRACT FROM AUTHOR]

An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion play a prominent role in the theory of n-tuple algebras of associative type. Our paper is devoted to the development of the variety theory of n-tuple semigroups. We construct a free k-nilpotent n-tuple semigroup and characterize the least k-nilpotent congruence on a free n-tuple semigroup. [ABSTRACT FROM AUTHOR]

Let o (G) be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between o (G) and o (H) , where H is a subgroup of G. Let N be the class of finite nilpotent groups and let L (G) be the subgroup lattice of G. In this paper, we show that the set { o (G) o (H) | G ∈ N , H ∈ L (G) } is dense in [ 0 , ∞) . Other density results are outlined throughout the paper. [ABSTRACT FROM AUTHOR]

Suppose that L is a Lie superalgebra over a field F of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of L is constructed and is proved to be exact. Moreover, the multiplier of L is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of L. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when L is nilpotent and finite-dimensional. [ABSTRACT FROM AUTHOR]

Let L and K be two Lie algebras over a commutative ring with identity. In this paper, under some conditions on L and K , it is proved that every triple homomorphism from L onto K is the sum of a homomorphism and an antihomomorphism from L into K. We also show that a finite-dimensional Lie algebra L over an algebraically closed field of characteristic zero is nilpotent of class at most 2 if and only if the sum of every homomorphism and every antihomomorphism on L is a triple homomorphism. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a fixed prime divisor of | G |. We prove that if every maximal subgroup of G is nilpotent, or normal, or has p ′ -order, then (1) G is solvable; (2) G has a Sylow tower; (3) there exists at most one prime divisor q ≠ p of | G | such that G is neither q -nilpotent nor q -closed. [ABSTRACT FROM AUTHOR]

Let K 〈 X 〉 denote the free associative algebra generated by a set X = { x 1 , ... , x n } over a field K of characteristic 0. Let I p , for p ≥ 2 , denote the two-sided ideal in K 〈 X 〉 generated by all commutators of the form [ u 1 , ... , u p ] , where u 1 , ... , u p ∈ K 〈 X 〉. We discuss the GL (n , K) -module structure of the quotient K 〈 X 〉 / I p + 1 for all p ≥ 1 under the standard diagonal action. We give a bound on the values of partitions λ such that the irreducible GL (n , K) -module V λ appears in the decomposition of K 〈 X 〉 / I p + 1 as a GL (n , K) -module. As an application, we take K = C and we consider the algebra of invariants (C 〈 X 〉 / I p + 1) G for G = SL (n , C) , O (n , C) , SO (n , C) , or Sp (2 s , C) (for n = 2 s). By a theorem of Domokos and Drensky, (C 〈 X 〉 / I p + 1) G is finitely generated. We give an upper bound on the degree of generators of (C 〈 X 〉 / I p + 1) G in a minimal generating set. In a similar way, we consider also the algebra of invariants (C 〈 X 〉 / I p + 1) G , where G = UT (n , C) , and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in C 〈 X 〉 G from the point of view of Classical Invariant Theory. In particular, for all G as above we give a criterion when a G -invariant of C 〈 X 〉 belongs to I p. [ABSTRACT FROM AUTHOR]

Let O(k) be the octonion Cayley-Dickson algebra over a commutative associative ring κ with 1. Let G(k) be the Moufang loop of invertible elements of O(k). Let H be a class of groups such that a group G is a member of H if and only if G satisfies the following three conditions: (a) G is not class-2 nilpotent. (b) G has a proper class-2 nilpotent subgroup. (c) G is not isomorphic to any subgroup of the group GL2(F) for any field F. The theorem proved in the paper states that if κ is an integral domain with 1+1 6= 0, then G(k) does not contain any subloop isomorphic to a group of class H, while if κ is an integral domain such that 1+1 = 0, then G(k) contains no subloop isomorphic to a class-2 nilpotent group at all. [ABSTRACT FROM AUTHOR]

Let G be a finite nilpotent group. We prove the following results. (1) If G is of class 2 and acts faithfully and irreducibly on an elementary abelian group V , then all nontrivial orbits of G on V have sizes larger than | G | 1 / 2 . (2) If G ′ is cyclic, then every subgroup of G intersecting trivially with the center of G has order less than | G | 1 / 2 . We also show that a result like (2) cannot be obtained when the hypothesis that G ′ is cyclic is replaced by the hypothesis that the center of G is cyclic. [ABSTRACT FROM AUTHOR]

Let be the class of all groups in which the normal closure of each -generated subgroup of belongs to . It is known that if is a quasivariety of groups then so is . We find the conditions on for the sequence to contain infinitely many different quasivarieties. In particular, such are the quasivarieties generated by a finitely generated nilpotent nonabelian group. [ABSTRACT FROM AUTHOR]

First, we prove the Kac--Wakimoto conjecture on modular invariance of characters of exceptional affine W-algebras. In fact more generally we prove modular invariance of characters of all lisse W-algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these W-algebras, which includes all exceptional W-algebras of type A and lisse subregular W-algebras in simply laced types. Third, for the latter cases we compute S-matrices and fusion rules. Our results provide the first examples of rational W-algebras associated with nonprincipal distinguished nilpotent elements, and the corresponding fusion rules are rather mysterious. [ABSTRACT FROM AUTHOR]

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc.361 (2009), 7, 3401–3434]. Later, this result was extended to all abelian groups [D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math.298 (2016), 612–653] and, recently, to all torsion finitely quasihamiltonian groups [A. Giordano Bruno and F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups, J. Group Theory23 (2020), 5, 831–846]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory20 (2017), 4, 763–774]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups. [ABSTRACT FROM AUTHOR]

We say a group 퐺 has property R ∞ if the number R ⁢ (φ) of twisted conjugacy classes is infinite for every automorphism 휑 of 퐺. For such groups, the R ∞ -nilpotency degree is the least integer 푐 such that G / γ c + 1 ⁢ (G) has property R ∞ . In this work, we compute the R ∞ -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n . Moreover, we compute the lower central series of Γ n , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ (Γ n) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c . [ABSTRACT FROM AUTHOR]

In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ (q) or Sz ⁡ (2 2 ⁢ e + 1) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ (q) or Sz ⁡ (2 2 ⁢ e + 1) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 푝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem. [ABSTRACT FROM AUTHOR]

Suppose that 풞 is a class of groups consisting only of periodic groups and P ⁢ (C) ′ is the set of prime numbers that do not divide the order of any element of a 풞-group. It is easy to see that if a subgroup 푌 of a group 푋 is 풞-separable in this group, then it is P ⁢ (C) ′ -isolated in 푋. Let us say that 푋 has the property C ⁢ - ⁢ S ⁢ e ⁢ p if all of its P ⁢ (C) ′ -isolated subgroups are 풞-separable. We find a condition that is sufficient for a nilpotent group 푁 to have the property C ⁢ - ⁢ S ⁢ e ⁢ p provided 풞 is a root class. We also prove that if 푁 is torsion-free, then the indicated condition is necessary for this group to have C ⁢ - ⁢ S ⁢ e ⁢ p . [ABSTRACT FROM AUTHOR]

Let G be a compact Hausdorff group. We show that if for every element x ∈ G there exists a positive integer q = q (x) such that xq is Engel, then G is locally virtually nilpotent. Furthermore, we show that if G is a finitely generated compact Hausdorff group in which any commutator [ x , y ] in G is Engel, then the commutator subgroup G ′ is locally nilpotent. [ABSTRACT FROM AUTHOR]