In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement and point out a version of 'relative' bounded generation results for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of quasisimple groups.
New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups $X_{n_i}(q)$ ($i\in I$) for $X\in\{{\rm PGL},{\rm PSp},{\rm PGO}^{(\varepsilon)},{\rm PGU}\}$ ($\varepsilon=\pm$) along an ultrafilter $\mathcal{U}$ on the index set $I$ for which $n_i\to_{\mathcal{U}}\infty$ determines the type $X$ and the field size $q$ up to the possible isomorphism of a metric ultraproduct of groups ${\rm PSp}_{n_i}(q)$ and a metric ultraproduct of groups ${\rm PGO}_{n_i}^{(\varepsilon)}(q)$. This extends results of Thom a…
In this paper, we shall prove that an ultraproduct of non-abelian finite simple groups is either finite simple, or has no finite dimensional unitary representation other than the trivial one. Then we shall generalize this result for other kinds of quasirandom groups. A group is called D- quasirandom if all of its nontrivial representations over the complex numbers have dimensions at least D. We shall study the question of whether a non-principal ultraproduct of a given sequence of quasirandom groups remains quasirandom, and whether an ultraproduct of increasingly quasirandom groups becomes min…
Groups that can be approximated by finite groups have been the center of much research. This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups? Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups? Certain cases were already well-studied. For example, if we have a pro-solvable group, then it is well-known that any finite abstract quotients are s…
If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then the normalizer of G in X is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic.
We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion.
Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we determine the minimal base size of $G$ when $E(G)/Z(E(G))$ is a finite simple group of Lie type in cross-characteristic. We show that $G$ has a regular orbit on $V$, with specific exceptions, for which we find the base size.
The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.
In the present paper we prove lemmata on strong contractibility in asymptotic cones and metric ultraproducts which we apply to both the case of finitely generated word norms and the case of conjugation invariant norms. We recover classically known contractibility results on free products and prove the contractibility of the asymptotic cone of the infinite symmetric group $\Sym_\infty$ equipped with a conjugation invariant norm. Furthermore, we give examples of contractible metric ultraproducts arising from subgroups of general linear groups. Additionally, we discuss algebraic properties of gro…
Motivated by the maximal subgroup problem of the finite classical groups we begin the classification of imprimitive irreducible modules of finite quasisimple groups. We obtain our strongest results for modules over fields of characteristic 0, although much of our analysis carries over into positive characteristic. If G is a finite quasisimple group of Lie type, and K an algebraically closed field of cross characteristic, we prove that an imprimitive irreducible KG-module is Harish-Chandra induced. We then specialize to the case when K has characteristic 0 and apply Harish-Chandra philosophy to…
In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of the classical socle and $q$ is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on class numbers, minimal …
We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hype…
A finite group $G$ is called $ψ$-divisible if $ψ(H)|ψ(G)$ for any subgroup $H$ of $G$, where $ψ(H)$ and $ψ(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $ψ$-divisible. Since the existence of $ψ$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $ψ$-divisibility property of ZM-groups. In the end, we introduce the concept of $ψ$-normal divisible group, i.e. a group for which the $ψ$-divisibility property…
The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3,4).
Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module $W$, chosen minimally with respect to containing the image of $H$ under the associated representation. We consider the question of when $H$ can act irreducibly on a $G$-constituent of $W^{\otimes e}$ and study its relationship to the maximal subgroup problem for finite classical groups.
Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as $\F_q$, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of exceptional Lie type over $\F_q$ which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from $G$ to the standard copy of $G(q)$. If $G \not\cong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.
We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.
Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $Π$-property in $G$ if there exists a chief series $\mathitΓ_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq i\leq n$) of $\mathitΓ_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ is a $π(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})$-number. Our main results are listed here: Theorem A. Let $\mathfrak{F}$ be a solubly saturated formation containing $\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\in \mathfrak{F}$. Let $X\unlhd G$ such that $F_p^*(E)\leq X\le…
Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spac…
Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ i…
In 1984, Michael Aschbacher proved a seminal classification theorem for the maximal subgroups of effectively all of the classical groups. In this thesis we give a comprehensive, yet accessible description and proof of Aschbacher's theorem, restricting its scope to the general linear group. The main theorem of this paper classifies the maximal subgroups of the general linear group into nine different classes; eight of which have natural descriptions based on an object that their members act on and stabilise, whilst the ninth class - though not having such a natural description - contains groups…
Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic gro…
