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In the present article we consider infinite locally finite perfect groups that are product of proper normal soluble subgroups of derived length at most n ( Π S n -groups for short) for some fixed positive integer n and give structural results for these groups. Also we provide some consequences for certain locally finite minimal non- X -groups for some classes X and barely transitive groups in the context mentioned above. Furthermore, we define certain collections of groups and consider their intersection with Π S n . Finally, we investigate some groups which have non-trivial minimal non- FC epimorphic images. [ABSTRACT FROM AUTHOR]

Let γn = [x1, . . ., xn] be the nth lower central word and Xn(G) the set of γn-values in a group G. Suppose that G is a profinite group where, for each g ∈ G, there exists a positive integer n = n(g) such that the set gXn(G) = {gy - y ∈ Xn(G)} contains less than 2N0 elements. We prove that G is a finite-by-nilpotent group. [ABSTRACT FROM AUTHOR]

In this paper we investigate some properties of the Burnside ring of a profinite group as defined in [5]. We introduce the notion of the crossed Burnside ring of a profinite FC-group, and generalise some results from finite to profinite (FC-)groups. In our investigations, we also obtain results on profinite FC-groups which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite d-generated group with high probability. [ABSTRACT FROM AUTHOR]

A group G is said to be an FC-group if every conjugacy class of G has finite order and is said to be a T-group if every subnormal subgroup is normal in G. In this paper we give a characterization of groups that are both FC-groups and T-groups and go further in the study of FC-groups with few subnormal non-normal subgroups in the sense of the chain conditions. The structure of FC-groups satisfying the maximal, the minimal and the double chain condition on subnormal non-normal subgroups will be described. [ABSTRACT FROM AUTHOR]

Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators. [ABSTRACT FROM AUTHOR]

A group G is said to have restricted centralizers if for each g in G the centralizer C G (g) either is finite or has finite index in G. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes π , we take interest in profinite groups with restricted centralizers of π -elements. It is shown that such a profinite group has an open subgroup of the form P × Q , where P is an abelian pro- π subgroup and Q is a pro- π ′ subgroup. This significantly strengthens a result from our earlier paper. [ABSTRACT FROM AUTHOR]

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we give a necessary and sufficient condition on a finite p-group G for the group AutΦ(G) to be elementary abelian. [ABSTRACT FROM AUTHOR]

By a result of Cocke and Venkataraman we know that if G is a group with at most m elements of maximal order, then | G | is m -bounded. In this paper we consider the following related setting. Suppose G is a group with at most m commutators whose order is maximal among all commutators. What can we say about the structure of the group G ? [ABSTRACT FROM AUTHOR]

Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup ⟨x,xα⟩ is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated.

Let n = [x1; : : : ; xn] be the nth lower central word. Suppose that G is a profinite group where the conjugacy classes xn(G) contains less than 2 @0 elements for any x 2 G. We prove that then n+1(G) has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group G is finite-by-nilpotent if and only if there is a positive integer n such that xn(G) contains less than 2 @0 elements, for any x 2 G. [ABSTRACT FROM AUTHOR]

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math.48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if G and H have same two-sided configuration sets and N is a normal subgroup of G with polycyclic or FC quotient, then there is a normal subgroup 𝔑 of H such that G / N ≅ H / 𝔑. Also, we show that if G and H are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic. [ABSTRACT FROM AUTHOR]

A group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite. [ABSTRACT FROM AUTHOR]

Let c be a positive integer. Suppose that each element of a group G has only a finite number of conjugates by γcG. If γcG is contained in the FC-center of G, then G is called an FCc-group; if there is a positive integer such that no element of γcG has more than d conjugates in G, then G is called a BFCc-group; if the index of C in G is finite with , then G is called a CFc-group. For FCc-group G, we study the finiteness conditions on the conjugate classes of abelian subgroups in γcG and establish a characterization of CFc-groups. A finite-by-(nilpotent of class c) is denoted by an FNc-group. Another aim of this article is to establish a characterization of FNc-group by means of the finiteness condition on normal closure of such type of subgroups. [ABSTRACT FROM AUTHOR]

Abstract Take a complete (sub-)critical Galton–Watson branching tree with finitely many leaves almost surely. Picking two distinct leaves at random, we ask for the height (as measured from the root) of their latest common ancestor. Upon conditioning on the first branching event we compute the distribution of this height, with calculations explicit in the case of a binary tree. [ABSTRACT FROM AUTHOR]

Let F, N, A and N2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN2-group. Moreover, G/Z2(G) is finite. [ABSTRACT FROM AUTHOR]

We analyze the complexity of the bi-embeddability relations for countable torsion-free abelian groups and for countable torsion abelian groups. [ABSTRACT FROM AUTHOR]

Abstract Let G be a group. The orbits of the natural action of Aut (G) on G are called "automorphism orbits" of G , and the number of automorphism orbits of G is denoted by ω (G). In this paper we prove that if G is an FC-group with finitely many automorphism orbits, then the derived subgroup G ′ is finite and G admits a decomposition G = Tor (G) × D , where Tor (G) is the torsion subgroup of G and D is a divisible characteristic subgroup of Z (G). We also show that if G is an infinite FC-group with ω (G) ⩽ 8 , then either G is soluble or G ≅ A 5 × H , where H is an infinite abelian group with ω (H) = 2. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits. [ABSTRACT FROM AUTHOR]

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations. [ABSTRACT FROM AUTHOR]

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra 458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order. [ABSTRACT FROM AUTHOR]

Abstract The (2 + 1) -dimensional Maxwell–Chern–Simons gauge model consisting of two complex scalar fields interacting through a common Abelian gauge field is considered. It is shown that the model has a solution that describes a soliton system consisting of vortex and Q-ball constituents. This two-dimensional soliton system possesses a quantized magnetic flux and a quantized electric charge. Moreover, the soliton system has a nonzero angular momentum. Properties of this vortex-Q-ball system are investigated by analytical and numerical methods. It is found that the system combines properties of a vortex and a Q-ball. [ABSTRACT FROM AUTHOR]

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here “almost” and “close” are in terms of the Hilbert–Schmidt norm. Using tracial 2-norm on I I 1 -factors we similarly define I I 1 -factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are I I 1 -factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group H 3 and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it. [ABSTRACT FROM AUTHOR]

For a finite group G, we denote by p(G) the minimal degree of faithful permutation representations of G, and denote by q(G) and c(G), the minimal degree of faithful representation of G by quasi-permutation matrices over the rational field ℚ and the complex field ℂ, respectively. In this paper, we calculate p(G), q(G) and c(G) for the groups of order p5, where p is an odd prime. [ABSTRACT FROM AUTHOR]

Using the technique of linear limits of characters due to Dade and Loukaki, we give some conditions on the supersolvable residual of a finite solvable group G that is sufficient to guarantee that G is an M-group. The monomiality of normal subgroups and Hall subgroups of the group G are also determined. [ABSTRACT FROM AUTHOR]

A group G is called an F Cc-group if each element of G has only finitely many conjugates by γcG and γcG lies in the F C-center of G. A group G is said to be a just non-minF Cc-group (a JNminF Cc-group) if all its proper quotients are minF Cc-groups, but G itself is not, where minF Cc-groups denote F Cc-groups with the minimal condition (min). The aim of this article is to give a complete description of the structure of JNminF Cc-groups. [ABSTRACT FROM AUTHOR]

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption V = L, that there is a first-order definable free abelian group with no first-order definable basis. [ABSTRACT FROM AUTHOR]

We consider several conjectures on abelian varieties over number fields whose common feature is a bound that depends only on the dimension of the variety and the degree of the number field. For example, Coleman's conjecture predicts that only a finite number of rings can occur as endomorphism rings once these two parameters are fixed. We show that this conjecture implies the existence of a small polarization as well as a uniform isogeny conjecture (without Faltings height) which in turn implies the uniform torsion conjecture. We then discuss several variants of the Lang-Silverman conjecture on heights and implications between them. In particular, we show how a rather weak version is, under Coleman's conjecture, equivalent to much more precise versions. We build on Bertrand's work to give a bound explicit in terms of the polarization. [ABSTRACT FROM AUTHOR]

We restrict irreducible characters of finite groups of degree divisible by p to their Sylow p-subgroups and study the number of linear constituents. [ABSTRACT FROM AUTHOR]

We derive some structural properties of a trifactorised finite group $G=AB=AC=BC$, where $A$, $B$, and $C$ are subgroups of $G$, provided that $A=A_{\unicode[STIX]{x1D70B}}\times A_{\unicode[STIX]{x1D70B}^{\prime }}$ and $B=B_{\unicode[STIX]{x1D70B}}\times B_{\unicode[STIX]{x1D70B}^{\prime }}$ are $\unicode[STIX]{x1D70B}$-decomposable groups, for a set of primes $\unicode[STIX]{x1D70B}$. [ABSTRACT FROM AUTHOR]

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S 3 . [ABSTRACT FROM AUTHOR]

The existence and uniqueness of linking systems associated to saturated fusion systems over discrete p -toral groups were proved by Levi and Libman. Their proof makes indirect use of the classification of the finite simple groups. Here we extend some results and arguments of Glauberman and Lynd to show how to remove this assumption. [ABSTRACT FROM AUTHOR]

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite. [ABSTRACT FROM AUTHOR]

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: -- The field k is p-special for some prime p different from its characteristic. -- The field k is perfect and fertile, and n = 3. [ABSTRACT FROM AUTHOR]

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6] ). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z p 5 for all odd primes p and Z p 4 for all odd primes p such that p ≡ 3 mod 4 . Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called an $${\mathcal{H}}$$ -subgroup in G if $${N_{G}(H)\cap H^{g} \leq H}$$ for all $${g\in G}$$ . A subgroup H of G is called weakly c-supplemented in G if G has a subgroup K such that G = HK and $${ H\cap K}$$ is an $${\mathcal{H}}$$ -subgroup in G. We investigate the structure of finite groups by means of weakly c-supplemented subgroups. Some recent results about supersolvability of finite groups are generalized to a saturated formation containing the class of all supersolvable groups. [ABSTRACT FROM AUTHOR]

We study free actions of compact groups on unital C ⁎ -algebras. In particular, we provide a complete classification theory of these actions for compact Abelian groups and explain its relation to the classification of classical principal bundles. [ABSTRACT FROM AUTHOR]

Let B be a 3-block of a finite group G with a defect group D. In this paper, we are mainly concerned with the number of characters in a particular block, so we shall use Isaacs' approach to block structure. We consider the block B of a group G as a union of two sets, namely a set of irreducible ordinary characters of G having cardinality k( B) and a set of irreducible Brauer characters of G having cardinality l( B). We calculate k( B) and l( B) provided that D is normal in G and $$D \cong \left\langle {x,y,z|x^{3^n } = y^{3^m } = z^3 = \left[ {x,z} \right] = \left[ {y,z} \right] = 1,\left[ {x,y} \right] = z} \right\rangle \left( {n > m \geqslant 2} \right)$$ . [ABSTRACT FROM AUTHOR]

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring , i.e. the prime graph of the normalized unit group of , coincides with that one of the group . In this note, we prove for finite groups a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups whose order is divisible by at most three primes and show that the Gruenberg-Kegel graph of such groups coincides with the prime graph of . [ABSTRACT FROM AUTHOR]

We classify abelian C -minimal valued groups up to pure and elementary extensions and small modifications . We define first the notion of almost regularity for abelian valued groups. Then we introduce on the chain of valuations the structure which enables us to characterize C -minimal abelian valued groups as the almost regular ones with an o-minimal chain. [ABSTRACT FROM AUTHOR]

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant characters for finite reflection groups, nilpotent groups and complete monomial groups. We also propose some conjectures about the structure of the groups admitting such characters. [ABSTRACT FROM AUTHOR]

For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) for it to be verbally closed. [ABSTRACT FROM AUTHOR]

For a finite groupG, let |Cent(G)| andω(G) denote the number of centralizers of its elements and the maximum size of a set of pairwise noncommuting elements of it, respectively. A groupGis calledn-centralizer if |Cent(G)| = nand primitiven-centralizer if. In this paper, among other results, we find |Cent(G)| andω(G) whenis minimal nonabelian and this generalizes some previous results. We give a necessary and sufficient condition for a primitiven-centralizer groupGwith the minimal nonabelian central factor. Also we show that if, thenGis a primitive 14-centralizer group andω(G) = 10 or 13. Finally we confirm Conjecture 2.4 in [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), 139-146]. [ABSTRACT FROM AUTHOR]

We present a systematic study of entire symmetric solutions $${u : \mathbb{R}^n \rightarrow\mathbb{R}^m}$$ of the vector Allen-Cahn equation where $${W:\mathbb{R}^m \rightarrow \mathbb{R}}$$ is smooth, symmetric, nonnegative with a finite number of zeros, and where $${ W_u= (\partial W / \partial u_1,\dots,\partial W / \partial u_m)^{\top}}$$ . We introduce a general notion of equivariance with respect to a homomorphism $${f:G\rightarrow\Gamma}$$ ( $${G, \Gamma}$$ reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen-Cahn equation and on a pointwise estimate for vector minimizers. [ABSTRACT FROM AUTHOR]

We establish, for finitely generated abelian semigroups G of matrices on Rn, and by using the extended limit sets (the J-sets), the following equivalence analogous to the complex case: (i) G is hypercyclic, (ii) JG(vn) = n for some vector n given by the structure of G, (iii) G(vn ) = Rn. This answer a question raised by the author. Moreover we construct for any n ≥ 2 an abelian semigroup G of GL(n;R) generated by n+1 diagonal matrices which is locally hypercyclic (or J-class) but not hypercyclic and such that JG(ek) = Rn for every k = 1, . . . , n, where (e1, . . . ,en) is the canonical basis of Rn. This gives a negative answer to a question raised by Costakis and Manoussos. [ABSTRACT FROM AUTHOR]

For a complex analytic variety with an action of a finite group and for an invariant 1-form on it, we give an equivariant version (with values in the Burnside ring of the group) of the local Euler obstruction of the 1-form and describe its relation with the equivariant radial index defined earlier. This leads to equivariant versions of the local Euler obstruction of a complex analytic space and of the global Euler obstruction. [ABSTRACT FROM AUTHOR]

Let X be a proper metric space and let F be a finite group acting on X by isometries. We show that the asymptotic dimension of F\X is the same as the asymptotic dimension of X. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and H B is a proper subgroup of G for every maximal subgroup H of H. The main aim of our investigation is to study the influence of ℳ-supplemented subgroups on the cyclic chief factors of finite groups. [ABSTRACT FROM AUTHOR]

Let be the prime graph associated with a finite group and be the degree pattern of . A finite group is said to be -fold -characterizable if there exist exactly nonisomorphic groups such that and . The purpose of this paper is two-fold. First, it shows that the symmetric group is -fold -charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, and , which are -fold -characterizable with . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the main results of Guo and Isaacs, Archiv der Mathematik 105 (2015) 215-222. We also obtain some criteria for -nilpotency of a finite group. [ABSTRACT FROM AUTHOR]

We give a sufficient condition for supersolubility of a finite group that is a product of two subnormal supersoluble subgroups. We prove that the supersoluble residual of such a group is equal to its nilpotent residual. Also we apply these results to finite groups that are a product of two subnormal p-supersoluble subgroups. [ABSTRACT FROM AUTHOR]