. 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.
