Classification Of Finite Groups Research Articles

Products of k atoms in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1, \ldots, u_k, v_1, \ldots , v_m\in H$ such that $u_1\cdot\ldots\cdot u_k=v_1\cdot\ldots\cdot v_m$. It is well-known that the sets $\mathcal U_k (H)$ are finite intervals whose maxima $\rho_k(H)=\max \mathcal U_k(H) $ depend only on $G$. If $|G|\le 2$, then $\rho_k (H) = k$ for every $k \in \mathbb N$. Suppose that $|G| \ge 3$. An elementary counting argument shows that $\rho_{2k}(H)=k\mathsf D(G)$ and $k\mathsf D(G)+1\le \rho_{2k+1}(H)\le k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ where $\mathsf D(G)$ is the Davenport constant. In \cite{Ga-Ge09b} it was proved that for cyclic groups we have $k\mathsf D(G)+1 = \rho_{2k+1}(H)$ for every $k \in \mathbb N$. In the present paper we show that (under a mild condition on the Davenport constant) for every noncyclic group there exists a $k^*\in \mathbb N$ such that $\rho_{2k+1}(H)= k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ for every $k\ge k^*$. This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in \cite{Ge-Gr-Yu15}.

Derangements in subspace actions of finite classical groups

This is the third in a series of four papers in which we prove a conjecture made by Boston et al. and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

Extremely primitive groups and linear spaces

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.

Classification of Finite Groups with Toroidal or Projective-Planar Permutability Graphs

Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph having all the proper subgroups of G as its vertices, and two subgroups are adjacent in Γ(G) if and only if they permute. In this article, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of K1, 5, P5, P6, C6, or K3, 3 as a subgraph.

Overgroups of root groups in classical groups

We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.

A Murnaghan–Nakayama rule for values of unipotent characters in classical groups

We derive a Murnaghan–Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai’s explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some ℓ \ell -singular element for certain primes ℓ \ell . As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes ℓ ≥ 3 \ell \ge 3 the first Cartan invariant in the principal ℓ \ell -block is larger than 2 unless Sylow ℓ \ell -subgroups are cyclic.

The Invariant Fields of the Sylow Groups of Classical Groups in the Natural Characteristic

Let X be any finite classical group defined over a finite field of characteristic p > 0. In this article, we determine the fields of rational invariants for the Sylow p-subgroups of X, acting on the natural module. In particular, we prove that these fields are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant linear forms defining X.

The set of minimal distances in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. If $G$ is finite, then there is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in \Delta^* (H)$, where $\Delta^* (H)$ denotes the set of minimal distances of $H$. We show that $\max \Delta^* (H) \le \max \{\exp (G)-2, \mathsf r (G)-1\}$ and that equality holds if every class of $G$ contains a prime divisor, which holds true for holomorphy rings in global fields.

On the modular composition factors of the Steinberg representation

Let G be a finite group of Lie type and Stk be the Steinberg representation of G, defined over a field k. We are interested in the case where k has prime characteristic ℓ and Stk is reducible. Tinberg has shown that the socle of Stk is always simple. We give a new proof of this result in terms of the Hecke algebra of G with respect to a Borel subgroup and show how to identify the simple socle of Stk among the principal series representations of G. Furthermore, we determine the composition length of Stk when G=GLn(q) or G is a finite classical group and ℓ is a so-called linear prime.

The simple classical groups of dimension less than 6 which are (2,3)-generated

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are PSL 3(q), q ≠ 4, and PSU 3(q2), q2 ≠ 9, 25. If n = 5 they are PSL 5(q), for all q, and PSU 5(q2), q2 ≥ 9. Also, the soluble group PSU 3(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups.

Arithmetic of seminormal weakly Krull monoids and domains

We study the arithmetic of seminormal v-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.

THE (2,3)-GENERATION OF THE CLASSICAL SIMPLE GROUPS OF DIMENSIONS 6 AND 7

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7.

Anomaly-safe discrete groups

We show that there is a class of finite groups, the so-called perfect groups, which cannot exhibit anomalies. This implies that all non-Abelian finite simple groups are anomaly-free. On the other hand, non-perfect groups generically suffer from anomalies. We present two different ways that allow one to understand these statements.

Infinite Groups of Finite Period

It is proved that there exist periodic groups containing elements of even order and only trivial normal 2-subgroups, in which every pair of involutions generates a 2-group. This gives a negative answer to Question 11.11a in the Kourovka Notebook. Furthermore, we point out examples of finite simple groups that are recognizable by spectrum in the class of finite groups but not recognizable in the class of all groups.

On the conjugacy separability of some free constructions of groups by root classes of finite groups

Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.

A Class of Finite Groups with Abelian Centralizer of an Element of Order 3 of Type (3, 2, 2)

In this work, we study the structure of finite groups in which the centralizer of an element of order 3 is isomorphic to Z3 × Z2 × Z2. The analysis is restricted to the class of groups whose order is not divisible by the prime number 5. It is shown that among finite simple groups such groups do not exist, and a detailed possible internal general structure of such groups is investigated. We use only those results that have been published before 1980.

On Rouquier blocks for finite classical groups at linear primes

W. Turner has proved that Broué's abelian defect group conjecture holds for certain unipotent blocks of the finite general linear group, the so-called Rouquier blocks. This together with theorems of A. Marcus and of J. Chuang and R. Rouquier proves that the conjecture holds for all blocks of such groups. We prove that other finite classical groups also possess analogues of Rouquier blocks at linear primes.

Harish-Chandra Series in Finite Unitary Groups and Crystal Graphs

The distribution of the unipotent modules (in non-defining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

A supercharacter theory for involutive algebra groups

If J is a finite-dimensional nilpotent algebra over a finite field k, the algebra group P=1+J admits a (standard) supercharacter theory as defined in [16]. If J is endowed with an involution σ, then σ naturally defines a group automorphism of P=1+J, and we may consider the fixed point subgroup CP(σ)={x∈P:σ(x)=x−1}. Assuming that k has odd characteristic p, we use the standard supercharacter theory for P to construct a supercharacter theory for CP(σ). In particular, we obtain a supercharacter theory for the Sylow p-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto in [7,8] for the special case of the symplectic and orthogonal groups.

Approximability of generalized free products of groups with amalgamated normal subgroup by some classes of finite groups

Let G be a free product of almost soluble groups A and B of finite rank with amalgamated normal subgroup H, where H ≠ A and H ≠ B, and let π be a finite set of primes. We prove that G is an almost residually finite π-group if and only if so are A, B, A/H, and B/H.