Volume 24, Issue 2

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group G of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group G in terms of its Lie rank and defining characteristic. When G is specified as G⁢(q){G(q)} and q is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of G .

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup Γ⁢(p){\Gamma(p)}, p a prime, is acted on by SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})}. Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})}. In this note, we compute the relation between these two spaces in the weight 2 case.

We construct a few supercharacter theories for finite semidirect products where the normal subgroup is of algebra group type. In the case of algebra groups, these supercharacter theories coincide with those of P. Diaconis and I. M. Isaacs. For the parabolic subgroups of GL⁢(n,𝔽q){\mathrm{GL}(n,\mathbb{F}_{q})}, the supercharacters and superclasses are classified.

Let G be a finite group. An irreducible character of G is called a 𝒫{\mathcal{P}}-character if it is an irreducible constituent of (1H)G{(1_{H})^{G}} for some maximal subgroup H of G . In this paper, we obtain some conditions for a solvable group G to be p -nilpotent or p -closed in terms of 𝒫{\mathcal{P}}-characters.

A subgroup H of a group G is said to be conditionally permutable (or c -permutable for short) in G if, for every subgroup T of G , there exists an element x∈G{x\in G} such that H⁢Tx=Tx⁢H{HT^{x}=T^{x}H}. A subgroup H of a group G is said to be completely c -permutable in G if, for every subgroup T of G , the subgroups H and T are c -permutable in 〈H,T〉{\langle H,T\rangle}. In this paper, we prove that H/HG{H/H_{G}} is nilpotent if H is a completely c -permutable subnormal subgroup of G . This result generalizes a well-known theorem of Ito and Szép, and gives a positive answer to an open problem in [W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015]. We also use complete c -permutability to determine the p -supersolubility of a group G .

We introduce two new notions of transitivity for Abelian p -groups based on isomorphism of quotients rather than the classical use of equality of height sequences associated with Abelian p -group theory. Unlike the classical theory where “most” groups are transitive, these new notions lead to much smaller classes, but even these classes are sufficiently large to be interesting.

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

We prove that the Fibonacci group F⁢(2,n){F(2,n)} for n odd and n≥11{n\geq 11} is hyperbolic. We do this by applying a curvature argument to an arbitrary van Kampen diagram of F⁢(2,n){F(2,n)} and show that it satisfies a linear isoperimetric inequality. It then follows that F⁢(2,n){F(2,n)} is hyperbolic.

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.