Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

In a recent paper, Frechette and Locus examined and found expressions for the infinite product [math] in terms of products of [math] -series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when [math] . These [math] -series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more [math] -series and their corresponding infinite products. Frechette and Locus left open the case where [math] . Here we find such an expression for the infinite products for [math] by making use of the new [math] -series obtained by Rains and Warnaar.

Cycle indices of linear, affine, and projective groups

Wronskian Appell polynomials and symmetric functions

Probability that the commutator of two group elements is equal to a given element

For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.

We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non-abelian matrix group remains open. For the purposes of polynomial time computation, we reduce this problem to the following “affine case:” Let A be an elementary abelian p-group which is a non-central minimal normal subgroup of a finite group G. Assume further that the quotient G/A is a finite simple group S of Lie type of characteristic p. The algorithmic goal is to distinguish the simple group S from the nonsimple group G. Both S and G are given as “black-box groups of characteristic p.” The reduction to this basic problem involves a large number of recent techniques and results which we review along the way. We address the affine case for the groups S = PSL(2, q) where q = p , p prime. We show that PSL(2, p) can be recognized in Monte Carlo polynomial time among all black-box groups of known characteristic. The situation for f ≥ 2 seems much harder; we exhibit challenging open cases for every f ≥ 2 when q is large. Along with S = PSL(2, p), the positive result holds for all simple groups S of Lie type of characteristic p with the property that in every nontrivial S-module in characteristic p, every element of S has a fixed point. We call these groups “unisingular.” Very recently, Guralnick, Saxl, and Tiep classified these groups. They found that a large number of classes of simple groups are unisingular, including certain classes of linear and unitary groups, all symplectic groups over an odd prime field, all orthogonal groups of odd degree over an odd prime field, many orthogonal groups of even degree, and a number of classes of exceptional groups. 2 Laszlo Babai and Aner Shalev

Generation and random generation: From simple groups to maximal subgroups

Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory (see e.g. [AKS, G+, LP, SS, V, WZ].) The explicit constructions of expander graphs [M1, M2, LPS] (see also [GG, Lu]) use deep mathematical tools to construct families of Cayley graphs of finite groups. The Independence Problem [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators. A counterexample to the general the problem was recently obtained in [ALW], by using a new combinatorial construction of expanders [RVW]. The problem remains for all classical series of finite simple groups (An, PSL(2,p), etc.)Let G be a finite group generated by at most d elements. The product replacement graph Γk(G) is defined to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. Most recently, graphs Γk(G) became a subject of an intense investigation, prompted by the study of a commonly used 'practical' product replacement algorithm for generating random elements in finite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on graphs Γk(G), showed a remarkable performance, as reported in [C+]. It was suggested in [LP], and proved in several special cases, that the product replacement graphs Γk(G) are expanders, for a fixed k, when |G| → ∞The main result of this paper is Theorem 2, establishing the connection between the expansion coefficient of the product replacement graph Γk(G) and the minimal expansion coefficient of a Cayley graph of group G with k generators. One can think of our result as of an additional evidence in favor of the speculation in [LP]. On the other hand, it gives an algorithmic motivation for study of the Independence Problem, in the aftermath of [ALW].In a special case (see Theorem 1 below), we show that if one assumes that all Cayley graphs with at most four generators in PSL (2, p) have a universal lower bound on expansion, then the product replacement graphs Γk(PSL(2,p)) form an expander family, when k ≥ 8 is fixed, and p → ∞.The proof of the Theorems consists of two parts: probability on groups and graph theoretic. The latter is based on a new combinatorial result of independent interest (Lemmas 4, 5), which generalizes a classical lemma that the direct product of expanders is an expander (see e.g. [Ch]). We prove that if a graph Γ on a direct product of sets satisfies two conditions:1) all 'row' subgraphs are expanders,2) at least (1-e) proportion of 'column' graphs are expanders,then graph &Gamma is also an expander. We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG, DS2, DS3, P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]).

Macdonald functions associated to complex reflection groups

Strongly generating elements in finite and profinite groups

Noncommutative symmetric functions and skewing operators

A famous problem in symmetric function theory is to find combinatorial formulas for Schur expansions of the Macdonald polynomials H ~ μ \tilde {H}_{\mu } . One such formula, valid for μ \mu satisfying μ 1 ≤ 3 \mu _1\leq 3 and μ 2 ≤ 2 \mu _2\leq 2 , involves Yamanouchi words weighted by Haglund’s statistics inv μ \operatorname {inv}_{\mu } and maj μ \operatorname {maj}_{\mu } . Previous proofs of this formula use the technical machinery of crystals and dual equivalence graphs. We give a new, elementary, and fully bijective proof of this formula based on the abacus model for antisymmetrized Macdonald polynomials. An extension to the Schur expansion of s ν H ~ μ s_{\nu }\tilde {H}_{\mu } is also provided.

Toeplitz Minors

Invariant geodetic problems on the affine group and related hamiltonian systems

In this paper, we first analyze the structure of a finite nonsolvable group in which every cyclic subgroup of order 2 and 4 of every second maximal subgroup is an NE-subgroup. Next, we prove that a finite group G is solvable if every nonnilpotent subgroup of G is a PE-group.