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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]

Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of dimension greater than or equal to two are the typical examples. It is conjectured that the only manifolds supporting an Anosov diffeomorphism are finitely covered by a nilmanifold, a type of manifold closely related to rational nilpotent Lie algebras. In this paper, we study the existence of Anosov diffeomorphisms for a large class of these nilpotent Lie algebras, namely the ones that can be realized as a rational form in a Lie algebra associated to a graph. From a given simple undirected graph, one can construct a complex c-step nilpotent Lie algebra, which in general contains different non-isomorphic rational forms, as described by the authors in previous work. We determine precisely which forms correspond to a nilmanifold admitting an Anosov diffeomorphism, leading to the first class of complex nilpotent Lie algebras having several non-isomorphic rational forms and for which all the ones that are Anosov are described. In doing so, we put a new perspective on certain classifications in low dimensions and correct a false result in the literature. [ABSTRACT FROM AUTHOR]

Let p be a prime and let G be a finite group such that the smallest prime that divides |G| is p. We find sharp bounds, depending on p, for the commuting probability and the average character degree to guarantee that G is nilpotent or supersolvable. [ABSTRACT FROM AUTHOR]

A well-known theorem of Baer states that in a given group G, the (n + 1) th term of the lower central series of G is finite when the index of the nth term of the upper central series is finite. Recently, Kurdachenko and Otal proved a similar statement for this theorem when the upper hypercenter factor of a locally generalized radical group has finite special rank. In this paper, we first decrease the Ellis' bound obtained for the order of γ n + 1 (G). Then we extend Kurdachenko's result for locally generalized radical groups. Moreover, some new upper bounds for the special rank of γ n + 1 (G , A) are also given, where A is a subgroup of automorphisms of G which contains inner automorphisms of G. [ABSTRACT FROM AUTHOR]

Motivated by the thesis work of Burkett, we here deduce some equivalent conditions for a supercharacter analog of nilpotent groups, S-nilpotent groups. Using isomorphism of supercharacters of groups, we also attempt to relate S-nilpotent groups and S-normal subgroups. Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

A nilmanifold is a quotient of a connected and simply connected nilpotent Lie group G by a uniform lattice N. In this paper, we determine the Reidemeister and Nielsen number of affine n-valued maps on such a nilmanifold. These are maps for which a given lifting to G splits into n affine maps of the Lie group G. To obtain this result we also establish a way of computing the number of generalized twisted conjugacy classes on finitely generated torsion-free nilpotent groups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank-n Carnot algebra is isomorphic to the exterior algebra of R n . Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces. [ABSTRACT FROM AUTHOR]

In this article, first, localizing at odd primes p, we study the self-homotopy groups of SU(4) and Sp(4). Second, we consider the homotopy nilpotency of Sp(4) when localized at odd prime p ≥ 7 . [ABSTRACT FROM AUTHOR]

We introduce the abstract concept of supernilpotence in loop theory, and relate it to existing concepts, namely, central nilpotence and nilpotence of the multiplication group. We prove that the class of supernilpotence is greater or equal than the class of nilpotence of the multiplication group, and combining existing results, we show that a finite loop is supernilpotent if and only if its multiplication group is nilpotent. We also provide a new exposition of a classical result and crucial ingredient, that loops with a nilpotent multiplication group are centrally nilpotent and admit a prime decomposition. [ABSTRACT FROM AUTHOR]

We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on 4-point subsets. We show that all δ -hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to ∞ . [ABSTRACT FROM AUTHOR]

We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we give a sharp bound for the size of the Schur multiplier of special p-group of all ranks. We also improve existing bounds for the size of Schur multiplier of p-groups (p ≠ 2 , 3) of class c ≥ 3 , and for p-groups of coclass r. [ABSTRACT FROM AUTHOR]

In this paper, we study how certain arithmetical conditions on the lengths of conjugacy classes of primary elements and π-elements of a finite group G influence the structure of G, some new results are obtained and several well-known results in the literature are developed. In particular, the structure of a kind of outer p-supersolvable groups is described, some new conditions implying π-supersolvability are found, and the upper bound for the order of Sylow subgroups is discussed. [ABSTRACT FROM AUTHOR]

In this paper we study the Reidemeister spectrum of finitely generated torsion-free 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds. [ABSTRACT FROM AUTHOR]

We prove a characterization of monomial projective representations of finitely generated nilpotent groups. We also characterize polycyclic groups whose projective representations are finite dimensional. [ABSTRACT FROM AUTHOR]

Mekler's construction gives an interpretation of any structure in a finiterelational language in a group (nilpotent of class $2$ and exponent $p>2$, butnot finitely generated in general). Even though this construction is not abi-interpretation, it is known to preserve some model-theoretic tamenessproperties of the original structure including stability and simplicity. Wedemonstrate that $k$-dependence of the theory is preserved, for all $k \in\mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtainfirst examples of strictly $k$-dependent groups (with no additional structure).

Motivated by a classic result for free groups, one says that a group G has the Magnus property if the following holds: Whenever two elements generate the same normal subgroup of G, they are conjugate or inverse-conjugate in G. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most 2. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free center-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most 2. On the way, we display 2-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property. [ABSTRACT FROM AUTHOR]

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010's. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in terms of sublevel sets of h-affine functions in step-2 rank-3 Carnot algebras that can be seen as a generalization of the one obtained by Cheeger and Kleiner in the Heisenberg setting. There is however a significant difference here as it is known that, unlike the Heisenberg setting, there are sublevel sets of h-affine functions on the free step-2 rank-3 Carnot algebra that are not half-spaces. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

We study some properties of the coset poset associated with the family of subgroups of class ≤ 2 of a nilpotent group of class ≤ 3 . We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of 4 × 4 upper unitriangular matrices over F p , and for the Burnside groups of exponent 3. [ABSTRACT FROM AUTHOR]

Let d(G) be the minimum number of elements required to generate a group G. For a group G of order p n with a derived subgroup of order p k and d (G) = d , we know the order of the Schur multiplier of G is bounded by p 1 2 (d - 1) (n - k + 2) + 1. In this paper, we find the structure of all p-groups that obtains the mentioned bound. Moreover, we show that all of them are capable. [ABSTRACT FROM AUTHOR]

Let G be a periodic group, and let LCM(G) be the set of all x ∈ G such that o (x n z) divides the least common multiple of o (x n) and o(z) for all z in G and all integers n. In this paper, we prove that the subgroup generated by LCM(G) is a locally nilpotent characteristic subgroup of G whenever G is a locally finite group. [ABSTRACT FROM AUTHOR]

A group G is said to be capable if it occurs as the central factor group H/Z(H) for some group H. Motivated by the results of Isaacs [11], in Proc. Amer. Math. Soc.129(10) (2001), pp. 2853-2859, we show that if G is a capable group with cyclic derived subgroup G′ of odd order, then |G/Z(G)| divides |(G/L)′|2ϕ(|L|)|L|, in which ϕ is Euler's function and L is the smallest term of the lower central series of G. Moreover, there is no such capable nonnilpotent group G that holds |G/Z(G)| = |G′|2. In particular, |G/Z(G)| = |G′|2 if and only if G is nilpotent. [ABSTRACT FROM AUTHOR]

Let U(n, ℚ) be the group of all n × n (upper) unitriangular matrices over rational numbers field ℚ. Let S be a subset of U(n, ℚ). In this paper, we prove that S is a subgroup of U(n, ℚ) if and only if the (i, j)-th entry Sij satisfies some condition (see Theorem 3.5). Furthermore, we compute the upper central series and the lower central series for S, and obtain the condition that the upper central series and the lower central series of S coincide. [ABSTRACT FROM AUTHOR]

We classify all finite groups with five relative commutativity degrees. Also, we give a partial answer to our previous conjecture on a lower bound of the number of relative commutativity degrees of finite groups. [ABSTRACT FROM AUTHOR]

We study the speed of convergence to the asymptotic cone for a finitely generated nilpotent group endowed with a word metric. The first result on this theme is given by Burago who showed that an abelian group endowed with a word metric converges to the normed space with the speed O 1 n in the sense of Gromov–Hausdorff distance. Later Krat showed the same statement for the Heisenberg group, and Breuillard and Le Donne constructed an example, the direct product of the Z and the Heisenberg group with a specific word metric, whose speed of convergence is precisely O 1 n . For 2-step nilpotent groups, we show that if the asymptotic cone is non-singular, then the speed of convergence is O 1 n for any choice of generating set. Our argument can be applied to every nilpotent Lie group with a left-invariant sub-Finsler metric. In terms of sub-Finsler geometry, the condition being non-singular is equivalent to the strongly bracket generating condition, and also to absence of abnormal curves. [ABSTRACT FROM AUTHOR]

In this short survey article, we aim to provide an up to date information on the progress made towards Schurs exponent property and related conjectures. We also mention the connection between Schurs exponent property and Noether's Rationality problem. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let ψ(G) denote the sum of element orders of G. It is well-known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group Cn. For nilpotent groups, we prove a natural generalization of this result, obtained by replacing the element orders of G with the element orders relative to a certain subgroup H of G. [ABSTRACT FROM AUTHOR]

Denote the sum of element orders in a finite group G by ψ (G) and let Cn denote the cyclic group of order n. In this paper, we prove that if | G | = n and ψ (G) > 13 21 ψ (C n) , then G is nilpotent. Moreover, we have ψ (G) = 13 21 ψ (C n) if and only if n = 6 m with (6 , m) = 1 and G ≅ S 3 × C m . Two interesting consequences of this result are also presented. [ABSTRACT FROM AUTHOR]

For a finite group $G$ , define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$ , where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$ , then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

Let G be a group and IA (G) denote the group of all automorphisms of G, which induce identity map on the abelianized group G ab = G / G ′ . Also the group of those IA -automorphisms which fix the centre element-wise is denoted by IA Z (G) . In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which IA Z -automorphisms are the same as central automorphisms, are either cyclic or elementary abelian. [ABSTRACT FROM AUTHOR]

For a free metabelian and nilpotent group G of finite rank n and class c, with n ≥ 2 and c ≥ 3 , we show that γ r ( IA G) ≠ I r + 1 A G for all 2 ≤ r ≤ c − 1 and we extend this result to several torsion-free nilpotent varieties of groups. By our methods, which are based on a connection of automorphisms of groups with automorphisms of Lie algebras, we also show the analogous result for several relatively free nilpotent Lie algebras. [ABSTRACT FROM AUTHOR]

Let α and β be automorphisms of a nilpotent p-group G of finite rank. Suppose that 〈(αβ(g))(βα(g))–1: g ∈G〉 is a finite cyclic subgroup of G, then, exclusively, one of the following statements holds for G and Γ, where Γ is the group generated by α and β. G is finite, then Γ is an extension of a p-group by an abelian group. G is infinite, then Γ is soluble and abelian-by-finite. [ABSTRACT FROM AUTHOR]