Volume 23, Issue 2

We classify finite non-solvable groups with a faithful primitive irreducible complex character that vanishes on a unique conjugacy class. Our results answer a question of Dixon and Rahnamai Barghi and suggest an extension of Burnside’s classical theorem on zeros of characters.

Let G be a solvable group, and let Δ⁢(G){\Delta(G)} be the character degree graph of G . In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that Δ⁢(G){\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and Δ⁢(G){\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field 𝔽{\mathbb{F}} with the property that there exists α∈𝔽{\alpha\in\mathbb{F}} such that M is similar to diag⁡(α⋅Idk,M1){\operatorname{diag}(\alpha\cdot\mathrm{Id}_{k},M_{1})}, where M1{M_{1}} is cyclic and 0≤k≤n{0\leq k\leq n}). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.

Here we note a minor variation on the method in [J. N. Bray, An improved method for generating the centralizer of an involution, Arch. Math. (Basel) 74 2000, 4, 241–245] which enables calculations of CH⁢(t){C_{H}(t)} for H a subgroup of a black box group G and t an involution of G . We then give some applications of this, the first to investigate the action of the Lyons sporadic group via conjugation on its involutions, a number of calculations in E6⁢(2){E_{6}(2)} and E8⁢(2){E_{8}(2)}, then finally to carry out some calculations in GL670⁢(2){\mathrm{GL}_{670}(2)}.

Fixing an arithmetic lattice Γ in an algebraic group G , the commensurability growth function assigns to each n the cardinality of the set of subgroups Δ with [Γ:Γ∩Δ][Δ:Γ∩Δ]=n{[\Gamma:\Gamma\cap\Delta][\Delta:\Gamma\cap\Delta]=n}. This growth function gives a new setting where methods of F. Grunewald, D. Segal and G. C. Smith’s “Subgroups of finite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that, for any unipotent algebraic ℤ{\mathbb{Z}}-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in p-s{p^{-s}}, where the degrees of the numerator and denominator are independent of p . This gives regularity results for the set of arithmetic lattices in G .

In this paper, under the usual stable range condition, a decomposition theorem for the elementary subgroup is obtained, and the injective stability theorem for odd-dimensional unitary K1{K_{1}} is proved.

We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

In this article, we study, in the context of complex representations of symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira in 2017. When applied to irreducible representations, this product gives rise to the Aguiar coefficients. We prove that these coefficients are in fact also branching coefficients for representations of connected complex reductive groups. This allows to use geometric methods already developed in a previous article, notably based on notions from geometric invariant theory, and to obtain some stability results on Aguiar coefficients, generalising some of the results concerning them given by Li Ying.

The study of the poset of hyperbolic structures on a group G was initiated by C. Abbott, S. Balasubramanya and D. Osin, [Hyperbolic structures on groups, Algebr. Geom. Topol. 19 2019, 4, 1747–1835, 10.2140/agt.2019.19.1747]. However, this poset is still very far from being understood, and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups wr⁡ℤnℤ{\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the lamplighter groups wr⁡ℤnℤ{\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} all have finitely many quasi-parabolic structures.