Volume 27, Issue 4

Let Ω be a set equipped with an equivalence relation ∼ \sim ; we refer to the equivalence classes as blocks of Ω. A permutation group G ≤ Sym ⁢ ( Ω ) G\leq\mathrm{Sym}(\Omega) is 𝑘-by-block-transitive if ∼ \sim is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article, we classify the finite block-faithful 2-by-block-transitive actions. We also show that, for k ≥ 3 k\geq 3 , there are no finite block-faithful 𝑘-by-block-transitive actions with nontrivial blocks.

It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.

Let 𝐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 𝖠. In this paper, we give an explicit description of the action of automorphisms of 𝐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 𝖠, J. Reine Angew. Math. 728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 𝐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 𝐺.

This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.

The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on Z m \mathbb{Z}_{m} from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual S ̂ \hat{S} of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on Z m \mathbb{Z}_{m} ; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of 𝑆, which coincides with the topological entropy of T = S ̂ T=\hat{S} by the so-called Bridge Theorem. In order to better understand and describe entropy, we introduce the degree deg ⁡ ( S ) \deg(S) and deg ⁡ ( T ) \deg(T) of 𝑆 and 𝑇.

We show that if 𝐺 is an admissible group acting geometrically on a CAT ⁡ ( 0 ) \operatorname{CAT}(0) space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary ( ∂ κ G , ν ) (\partial_{\kappa}G,\nu) is a model for the Poisson boundary of ( G , μ ) (G,\mu) , where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.