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In the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type F 4 . We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system CHEVIE, we spotted an error in Spaltenstein's tables for the generalised Springer correspondence in type E 7 . [ABSTRACT FROM AUTHOR]

For any prime p and S a p -group isomorphic to a Sylow p -subgroup of G 2 (p n) or PSU 4 (p n) with n ∈ N , we determine all saturated fusion systems supported on S up to isomorphism. [ABSTRACT FROM AUTHOR]

The structure of the nonsolvable (P)-groups is completely described in this article. By definition, a finite group G is called a (P)-group if any two cyclic p-subgroups of the same order are conjugate in G, whenever p is a prime number dividing the order of G. [ABSTRACT FROM AUTHOR]

The McKay–Navarro conjecture is a refinement of the McKay conjecture that additionally takes the action of some Galois automorphisms into account. We verify the inductive McKay–Navarro condition in the defining characteristic for the finite groups of Lie type with exceptional graph automorphisms, the Suzuki and Ree groups, B n (2) (n ≥ 2), and the groups of Lie type with non-generic Schur multiplier. This completes the verification of the inductive McKay–Navarro condition for the finite groups of Lie type in their defining characteristic. [ABSTRACT FROM AUTHOR]

The article presents the discussion on cyclicity, commutativity, nilpotence, and solvability.

We determine, for p odd, all saturated fusion systems F on a Sylow p -subgroup S of the unitary group S U 4 (p) and we prove that they are all realizable by finite groups. In particular, we prove that S does not support any exotic fusion systems. [ABSTRACT FROM AUTHOR]

Let G ≤ 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. [ABSTRACT FROM AUTHOR]

In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type in their defining characteristic. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters presents in the McKay conjecture [12]. Our proof of this case of the conjecture relies on a character correspondence constructed by Maslowski in [11]. Building on this we verify the inductive condition for Navarro's refinement from [14] for most groups of Lie type in defining characteristic. [ABSTRACT FROM AUTHOR]

We describe two methods for computing p -modular Brauer character tables for groups of Lie type G (p f) in defining characteristic p , assuming that the ordinary character table of G (p f) is known, and the weight multiplicities of the corresponding algebraic group G are known for p -restricted highest weights. [ABSTRACT FROM AUTHOR]

We prove that the cohomology group of a Deligne–Lusztig variety defines a Morita equivalence in a case which is not covered by the argument in [2] , specifically we consider the situation for semisimple elements in type D whose centralizer has non-cyclic component group. Some arguments use considerations already present in an unpublished note by Bonnafé, Dat and Rouquier. [ABSTRACT FROM AUTHOR]

For S a Sylow p-subgroup of the group G2(p) for p odd, up to isomorphism of fusion systems, we determine all saturated fusion systems F on S with Op(F)=1. For p≠7, all such fusion systems are realized by finite groups whereas for p=7 there are 29 saturated fusion systems of which 27 are exotic. [ABSTRACT FROM AUTHOR]

Abstract In 2006, C. Bonnafé posed a general question on characters of finite groups. A positive answer would have reduced drastically some proofs by G. Lusztig. [ABSTRACT FROM AUTHOR]

An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characteristic. [ABSTRACT FROM AUTHOR]

We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic groups and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. [ABSTRACT FROM AUTHOR]

The spectrum ω ( G ) of a finite group G is the set of element orders of G . Finite groups G and H are isospectral if their spectra coincide. Suppose that L is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic p . It is proved that a finite group G isospectral to L cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from p . Together with a series of previous results this implies that every finite group G isospectral to L is ‘close’ to L . Namely, if L is a linear or unitary group, then L ⩽ G ⩽ Aut L , in particular, there are only finitely many such groups G for given L . If L is a symplectic or orthogonal group, then G has a unique nonabelian composition factor S and, for given L , there are at most 3 variants for S (including S ≃ L ). [ABSTRACT FROM AUTHOR]

A finite group L is said to be quasisimple if L is perfect and L/ Z( L) is nonabelian simple, in which case we also say that L is a cover of L/Z( L). It has been proved recently (Nguyen, Israel J Math, ) that a quasisimple classical group L is uniquely determined up to isomorphism by the structure of ${{\mathbb C}} L$, the complex group algebra of L, when L/Z( L) is not isomorphic to PSL(4) or PSU(3). In this paper, we establish the similar result for these two open cases and also for covers with nontrivial center of simple groups of exceptional Lie type and sporadic groups. Together with the main results of Tong-Viet (Monatsh Math 166(3-4):559-577, , Algebr Represent Theor 15:379-389, ), we obtain that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of its complex group algebra. [ABSTRACT FROM AUTHOR]

Abstract: The main theorem is a step in a program to show there exist finite lattices that are not an interval in the lattice of subgroups of any finite group. As part of the proof of the main theorem, we prove a theorem on the structure of maximal parabolics in finite groups of Lie type, which is of independent interest. [Copyright &y& Elsevier]

Abstract: Important subalgebras of a Lie algebra of an algebraic group are its toral subalgebras, or equivalently (over fields of characteristic 0) its Cartan subalgebras. Of great importance among these are ones that are split: their action on the Lie algebra splits completely over the field of definition. While algorithms to compute split maximal toral subalgebras exist and have been implemented (Ryba, 2007; Cohen and Murray, 2009), these algorithms fail when the Lie algebra is defined over a field of characteristic 2 or 3. We present heuristic algorithms that, given a reductive Lie algebra L over a finite field of characteristic 2 or 3, find a split maximal toral subalgebra of L. Together with earlier work (Cohen and Roozemond, 2009) these algorithms are very useful for the recognition of reductive Lie algebras over such fields. [Copyright &y& Elsevier]

The article discusses a paper that aims at using methods from homotopy theory to prove that some pairs of fusion systems of finite groups of Lie type are isotypically equivalent. It gives the formulations for the authors' main theorem resulting from the methods followed, comparing the fusion of groups of Lie type with different defining characteristics. Also included are a few proven cases of isotypical equivalences between fusion systems of classical groups.

Abstract: We find a method to recognize the characteristic of a simple group of Lie type G from its Dirichlet polynomial . This is enough to complete the proof of the following statement: if G is a simple group, H is a finite group and , then . [Copyright &y& Elsevier]

Abstract: We investigate the complexity of constructing involutions and their centralisers in groups of Lie type over finite fields of odd order, and discuss applications to the problem of deciding whether a matrix group, or a black-box group of known characteristic, is simple. We show that if the characteristic is odd, then simplicity can be recognised in Monte Carlo polynomial time. [Copyright &y& Elsevier]

We give a new elementary construction of Ree's family of finite simple groups of type 2G2, which avoids the need for the machinery of Lie algebras and algebraic groups. We prove that the groups we construct are simple of order q3(q3 +1)(q-1) and act doubly transitively on an explicit set of q3 + 1 points, where q = 32k+1. Moreover, our construction is practical in the sense that generators for the groups and many of their maximal subgroups may easily be obtained. [ABSTRACT FROM AUTHOR]

A new elementary construction of the small Ree groups is described. [ABSTRACT FROM AUTHOR]

Abstract: We give an elementary construction of Ree''s family of finite simple groups , avoiding the need for the machinery of Lie algebras, algebraic groups, or buildings. We calculate the group orders and prove simplicity from first principles. Moreover, this is a practical construction in the sense that it gives an explicit description of the generalized octagon, and from it generators for many of the maximal subgroups may be easily obtained. [Copyright &y& Elsevier]

Abstract: Let L be the Lie algebra of a simple algebraic group defined over a field and let H be a split maximal toral subalgebra of L. Then L has a Chevalley basis with respect to H. If , it is known how to find it. In this paper, we treat the remaining two characteristics. To this end, we present a few new methods, implemented in Magma, which vary from the computation of centralizers of one root space in another to the computation of a specific part of the Lie algebra of derivations of L. [Copyright &y& Elsevier]

Abstract: Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon , and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group. [Copyright &y& Elsevier]

Abstract: Both Timmesfeld''s abstract root subgroups and simple Lie algebras generated by extremal elements lead to root filtration spaces: synthetically defined geometries on points and lines which can be characterized as root shadow spaces of buildings. Here we show how to obtain the root filtration space axioms from root subgroups and classical Lie algebras. [Copyright &y& Elsevier]

Let F be the class of finite groups which contain nonconjugate Carter subgroups and assume that F is nonempty. Then it has been shown that a member of F, of minimal order, should be an almost simple group A. In this paper, using CFSG, we restrict the possibilities for such A: in particular we show that A cannot be simple, except possibly when it is a unitary group. [Copyright &y& Elsevier]

Dedicated to the memory of Professor A. I. KostrikinThe main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations φ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations φ that can be realized over an unramified extension of Qp, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and G≠ G2(5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1