Residually finite groups with uniformly almost flat quotients

The general problem, with a particular instance of which the present paper is concerned, is to obtain a description of the local structure of a group from information about the global structure. The aspect of local structure investigated here is the embedding of subgroups, especially of nilpotent subgroups in finite soluble groups. A classification of embeddings of subgroups in finite groups by means of an arithmetic function called abnormal depth was proposed in [6]. Let H be a subgroup of a finite group G. Then a(G:H), the abnormal depth of H in G, is the least number of abnormal links appearing in any balanced chain of subgroups connecting H to G, that is a chain for which each link is either normal or abnormal. Thus a (G:H)= 0 if and only if H is subnormal in G; and a(G:P)__< 1 for every subgroup P of G of prime power order. It was shown in [6] that if H is a nilpotent subgroup of a finite soluble group G, of nilpotent length n, then a (G: H) =< n - 1. Here in w 1 we examine in greater detail the easiest non-trivial case, in which n = 2, and then in w 2 prove certain supplementary results for n = 3 and n = 4. Some simple wreath product properties are established in w 3 and used in w 4 for the construction of examples showing that the embedding results obtained cannot be improved in various obvious ways. Notation and terminology follow common usage. If t; and ~ are classes of groups, then 3s ~ denotes the class of all groups G having a normal subgroup X such that X e 3~ and G/X e ~. This defines a composition of classes of groups which in general is not associative. However, we shall deal only with classes of which the composition is associatNe, and we may therefore omit brackets from products of more than two classes. Since we shall be concerned exclusively with finite groups, we take 91 to denote the class of finite nilpotent groups and 9.1 the class of finite abelian groups. Then for any positive integer n, 9l" is the class of finite soluble groups of nilpotent lengths <__ n; and 9.I" is the class of finite soluble groups of derived lengths __< n. Henceforth the term group is understood to mean finite group. Then any group G has a unique smallest normal subgroup L such that G/L is nilpotent: G/L is called the 91-residual ofG. IfH is any subgroup of G, then there is a unique smallest normal subgroup of G containing H, called the normal closure of H in G and denoted by Ha; and a unique smallest subnormal subgroup of G containing H, called the subnormal closure of H in G and (following Wielandt [8]) denoted by H'" a. If H a = G, we shall say that H is contranormal in G. Then, for any subgroup H of G, it is clear that H is contranormal in H'" a. (This is to be compared with the fact that the hypernormalizer NE(H ) of H in G is self-normalizing in G.) An abnormal subgroup is both self-normalizing and

Variants of some of the Brauer-Fowler theorems

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset$\mathcal{C}$stable under inversion. We study the associated Killing formK. For the universal calculus associated to$\mathcal{C}$=G\ {e} we show that the magnitude$\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$of the Killing form is defined for all finite groups (even whenKis not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible,iffμ ≠ 1/(N− 1) whereNis the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N− 1)-dimensional subspace of invariant vectors is invertibleiffthe finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in anySnhas invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groupsD2nwithnodd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

The classification of the finite simple groups was completed sometime during the summer of 1980. To the extent that I can reconstruct things, the last piece in the puzzle was filled in by Ronald Solomon of Ohio State University. At the other chronological extreme, the theory of finite groups can be traced back to its beginnings in the early nineteenth century in the work of Abel, Cauchy, and Galois. Hence the problem of classifying the finite simple groups has a history of over a century and a half. The proof of the Classification Theorem is made up of thousands of pages in various mathematical journals with at least another thousand pages still left to appear in print. Many mathematicians have contributed to the proof; some have spent their entire mathematical lives working on the problem. The problem itself is one of the most natural in mathematics: the group is one of the fundamental structures of modern mathematics; the finite groups are a natural subclass of the class of all groups. Moreover, the finite group theorist is quickly led to consider simple groups via the composition series of a group, and if he is optimistic, to the hope that the finite simple groups might be determined explicitly and much of the structure of the arbitrary finite group retrieved from that of its composition factors. Despite all of this, and despite the fact that most mathematicians learn this much group theory before receiving their Ph.D., the average mathematician does not seem to known much about the classification problem or the mathematics developed to solve it. Within the obvious space limitations of this article, I hope to convey some idea of how the finite simple groups are classified and to relate some of the history of the effort. A more complete description appears in [6], while a very detailed two volume account (by Daniel Gorenstein) is in preparation. A preliminary version of the first quarter of Gorenstein's work appears in [19]. The proceedings of two recent conferences on simple groups containing expository articles on the classification will soon appear in [12] and [13]. Finally an article by Walter Feit on the history of finite group theory through 1961 will appear in [14]. I have included a reasonably lengthy bibliography. Still, many important papers are omitted as they are not directly encountered in the brief outline provided. Other fundamental papers have yet to appear. More complete bibliographies are contained in some of the books mentioned above. Section 1. The Finite Simple Groups

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

The irreducible characters of finite groups are always contained in blocks of defects which are nonnegative integers. Even though blocks always exist in finite groups, it is not the case that blocks of defect zero would always exist as well. Blocks of defect zero contain only one irreducible ordinary character each of defect zero and the defect group of blocks of defect zero is always the trivial subgroup of a finite group. Some finite groups do not have characters of defect zero and hence no blocks of defect zero either. The object in this paper is to study finite groups containing no blocks of defect zero. Finite abelian groups and p-groups will serve as special cases in this regard, with all blocks of finite abelian groups being of full/highest defect. We shall also determine an upper bound for the number of blocks in finite groups which contain no blocks of defect zero.

Finite quotients of ultraproducts of finite perfect groups

Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian [Formula: see text]-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively.

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

Certain finite linear groups of prime degree

For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G cong S times Z(G)$‎.

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

We say that a class of finite structures for a finite first-order signature is r-compressible for an unbounded function r: N → N+ if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log3-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is log3-compressible. The results rely on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology. We also indicate why the results are close to optimal.

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that $\mathbb{P}^2/A_5$ is not expressible as a quotient of a smooth variety by a finite abelian group.

The main result of this paper is that the outer automorphism group of a free product of finite groups and cyclic groups is semistable at infinity (provided it is one ended) or semistable at each end. In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. We also prove that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism groups of the free product of four finite groups or the free product of a single finite group with a free group of rank two are virtual duality groups of dimension two, in contrast with the above example. Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krstić and Vogtmann.