. Baer's theorem is one of the cornerstone result in group theory, providing critical insights into the relationship between the finiteness of central factor group and that of the commutator subgroup. Building upon Schur's foundational work, Baer's theorem connects the upper and lower central series, establishing constraints on group structure that have far-reaching implications. This paper provides a brief review of Baer's theorem, detailing its historical development, generalizations, and recent extensions. Some key results include exponents, bounds on central series, extensions to locally generalized radical groups, finite rank conditions and applications to automorphism-influenced properties are given. Invoking the notion of variety of groups, we also propound the Baer's (or Schur's) theorem in its most general form as a fundamental question and attempt to identify all classes of groups that are Schur-Baer with respect to some variety as potential answers. Particular attention is also given to some of its applications in diverse areas of mathematics. Furthermore, the paper explores open problems and potential research directions, underscoring the theorem's enduring significance and its role in shaping contemporary mathematical inquiry.
. For a nontrivial finite group G, the intersection graph Gamma(G) of G is a simple undirected graph whose vertices are the nontrivial proper subgroups of G and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper, we obtain the clique number of the intersection graph of cyclic groups whose orders have four prime divisors. Moreover we find the clique number of the intersection graph of cyclic groups of order n such that all powers of prime divisors of n are equal. As a special case, we find the clique number of this graph for the cyclic groups of the square-free orders.
. It is known that if G is a finite group in which all the elements of prime power order commute, then G is abelian. However, the same does not hold if prime power is replaced by prime. In this article, we introduce the study of a class of finite groups G in which the prime order elements commute. In particular, we discuss the relationship between these class of groups with other known classes of finite groups, like simple groups, perfect groups etc. Moreover, we also prove some results on the possible orders of such groups. Finally, we conclude with some open issues and observations supported by computational evidences using GAP.
A group G is said to satisfy the infinite trivial intersection property (ITIP for short), if for every pair of finite subgroups U, V such that U boolean AND V = 1, there exist infinite subgroups X and Y of G such U <= X and V <= Y and X boolean AND Y = 1. We shall say that a group G satisfies the infinite non-trivial intersection property (INIP) if every pair of infinite subgroups of G intersect non-trivially. The subject of this paper is to find classes of groups that satisfy ITIP. We prove, among other things, that every periodic locally nilpotent non-Chernikov group satisfies ITIP. The Pru & uml;fer-by-finite p-groups are examples of locally nilpotent Chernikov groups that do not satisfy ITIP. We then characterize locally nilpotent groups that satisfy INIP and structure theorems are given in the periodic and the non-periodic case.
. Factorizations play an important role in the analysis of the structure of finite groups. In this short note we extend a basic p-lo cal Factorization Theorem in the textbook The Theory of Finite Groups by H. Kurzweil and B. Stellmacher that is applied in their Signalizer Functor Chapter. Our result is somewhat surprising and immediately implies their Factorization Theorem and should have other important applications. The proof of this basic result is tricky and elementary and could have been proved several years ago. The proof requires only well known facts about finite group factorizations and Sylow p-subgruops of finite groups. It resulted from the goal of simplifying the proof of the p-Lo cal Factorization Theorem in the excellent H. Kurzweil and B. Stellmacher textbook Theory of Finite Groups. Hopefully this elementary result will inspire and lead to new important results on the structure of finite proofs, as is illustrated by the application to the Signalizer Functor Chapter in the textbook.
The nilpotent graph of a finite group G, denoted by Gamma(N)(G), is a simple graph whose vertex set is G nil(G), where nil(G) = {g is an element of G : < g, h > is nilpotent for all h is an element of G}, and two distinct vertices are related if they generate a nilpotent subgroup of G. In this work, lower bounds for the clique number and the number of connected components of Gamma(N)(G) are presented in terms of the size of its Fitting subgroup and the number of its strongly self-centralizing subgroups of G, respectively. We prove that no finite non-nilpotent group has a self-complementary nilpotent graph. Furthermore, for the dihedral group D-n, it is determined that the number of connected components of its nilpotent graph is one more than n when n is odd or one more than the 2 ' part of n when n is even. In addition, a formula for the number of connected components of Gamma(N)(PSL(2, q)), where q is a prime power, is provided.
Let the Ducci function, D, be an endomorphism on Znm such that D(x1, x2, ... , xn) = (x1 + x2 mod m, x2 + x3 mod m, ... , xn + x1 mod m). The sequence {D alpha(u)}infinity alpha=0 is the Ducci sequence of u for u E Znm. Because Znmis finite, the Ducci sequence of u enters a cycle for all u E Znm, which we call the Ducci cycle of u. In this paper, our main goal is to prove that if n is odd and m = 2lm1 where m1 is odd, then the longest it will take for a Ducci sequence in Znm to enter its cycle is l iterations of D. In addition to this, we will prove that the set of all tuples in Znm in a Ducci cycle for some u E Znm is {(x1, x2,. . . , xn) E Znm x1+x2+& centerdot;& centerdot;& centerdot;+xn-0 mod 2l}.
Since the classification theorem of finite simple groups was declared proved in the early 1980s, many group theorists have been attempting to delve deeper into the structure of simple groups from the perspective of group invariants, resulting in a series of research topics on the quantitative characterization of simple groups, such as spectral characterization, two-order characterization, and OD-characterization. In 2018, Moreto proposed a new conjecture for the characterization of finite simple groups by the group order and the number of elements of the largest prime order. A simple group whose order is divisible by exactly three distinct prime numbers is called a simple K3-group. These groups form a simple class of finite non-abelian simple groups. This paper establishes a characterization of L-2(8) and L(3()3) by combining the group order with the number of elements of the largest prime order, which shows that the conjecture holds for all simple K-3-groups except L-2(7), U-3(3) and U-4(2). In addition, we also characterize L-2(7), U-3(3) and U-4(2) under additional condition of non-solvability. Furthermore, we prove that a conjecture of Li and Shi holds for the alternating groups A8, A10, and L(2)7). Thus the conjecture of Li and Shi is valid for sporadic simple groups, for alternating groups An(n >= 5), and for all simple K3-groups except U-3(3) and U-4(2).
This paper examines the probability of finite groups being Hamiltonian,a property defined by all subgroups being normal, and its implications for group structure analysis. To this end, we introduce the Hamiltonian degree, a novel extension of commutativity degrees in finite groups, and propose a comprehensive framework for its evaluation. Explicit formulas are derived for the Hamiltonian degree of dihedral groups, alongside general bounds applicable to broader group classes. Additionally, we explore its relationship to conjugacy class subgroups, shedding light on new structural connections within group theory.
In this paper we focus on problems of irreducible p-Brauer characters and state conjectures based on many examples which we computed. Many of the asked questions hold true for p-solvable groups, but their answers in general seem to require a much deeper understanding than we have at the moment. The questions are dealing with degrees, Hilbert divisors, divisibility, height-zero irreducible Brauer characters, number of irreducible Brauer characters in a p-block, and Cartan invariants. For instance, one of the main conjectures states that an irreducible Brauer character has Hilbert divisor 1 if and only if the character lies in a p-block of defect zero. This is true for p-solvable groups since in this case there is a relation between Hilbert divisors and vertices. However, even for a p-block of a non-p-solvable group containing only two irreducible Brauer characters we do not have an idea how to attack the problem. We hope that the conjectures and questions which we state in this paper will inspire further research.
. This work is motivated by results obtained and problems posed by Bianchi, Camina, Lewis, Pacifici and Sanus, counting conjugacy classes of non-self-normalising subgroups of finite groups. We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups G = PSL2(p), p prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds 17, 18, 6 and 12 respectively satisfied by these invariants for all p > 37. A computer search carried out for a different but related problem shows that these bounds are attained for over a million primes p; we show that if the Bateman-Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes p.
The deep commuting graph of a given group G is a graph whose vertex set is G, and two elements of G are adjacent if their inverse images in every central extension of G commute. In this paper, we introduce the concept of deep isoclinism for groups and tie this concept to the deep commuting graphs.
The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. We address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Holt and Plesken's list of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP. We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this is the smallest order of such a group. This study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.
. Rota-Baxter operators on groups were introduced by L. Guo, H. Lang, Yu. Sheng in 2020. In 2023, V. Bardakov and the second author showed that all Rota-Baxter operators on simple sporadic groups are splitting, i. e. they correspond to exact factorizations of groups. In 2024, the authors of the current paper described all non-splitting Rota-Baxter operators on alternating groups. Now we describe Rota-Baxter operators on finite simple exceptional groups of Lie type and projective special linear groups of degree two.
We determine the finite groups G in which every subset A subset of G of cardinality dividing the order of G is a factor, i.e. has a complement B subset of G of cardinality G/A such that G = A & centerdot; B or G=B & centerdot; A.
A group G is said to have dense CD-subgroups if each non-empty open interval of the subgroup lattice L(G) contains a subgroup in the Chermak-Delgado lattice CD(G). In this note, we study finite groups satisfying this property.
Let G be a finite group and x be an element of G. Define SolG(x) as the set of all y is an element of G such that < x, y > is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely |N-G(< x >)| | | Sol(G)(x)| , where N-G(< x >) is the normalizer of < x >. Furthermore, we demonstrate that the conjecture holds in the special case where N-G(< x >) is a Frobenius group with kernel C-G(x), the centralizer of x and N-G(< x >) : C-G(x)| is of prime order. Finally, we will classify all finite simple groups G that contain an element x for which Sol(G)(x) is a maximal subgroup of order pq, where p and q are prime numbers.
We generalise the retractions to standard parabolic subgroups for even Artin groups to FC-type Artin groups and other more general families. We prove that these retractions uniquely extend to any parabolic subgroup. We use retractions to generalise the results of Antol & imath;n and Foniqi that reduce the problem of intersection of parabolic subgroups to weaker conditions. As a corollary, we characterise coherence for the FC case.
We study finite subgroups of outer automorphisms of free products. We give upper bounds for the orders of these finite subgroups as well as bounds for the orders of individual torsion outer automorphisms under some (necessary) conditions for the free factors.
Recently, sub-indices and sub-factors of groups with connections to number theory, additive combinatorics, and factorization of groups have been introduced and studied. Since all group subsets are considered in the theory and there are many basic open problems, conjectures, and questions, their computational aspects are particularly important. In this paper, by introducing some computational methods and using theoretical approaches together, we not only solve several problems but also pave the way to study the topic. As the most important result of the study, we completely characterize finite index stable groups.
