Volume 28, Issue 6

We show that a generically sharply 𝑡-transitive permutation group of finite Morley rank on a set of rank 𝑟 satisfies t ≤ r + 2 t\leq r+2 provided the pointwise stabilizer of a generic ( t − 1 ) (t-1) -tuple is an 𝐿-group, which holds, for example, when this stabilizer is solvable or when r ≤ 5 r\leq 5 . This makes progress towards establishing the natural bound on 𝑡 implied by the Borovik–Cherlin conjecture that every generically ( r + 2 ) (r+2) -transitive permutation group of finite Morley rank on a set of rank 𝑟 is of the form PGL r + 1 ⁡ ( F ) \operatorname{PGL}_{r+1}(F) acting naturally on P r ⁢ ( F ) \mathbb{P}^{r}(F) . Our proof is assembled from three key ingredients that are independent of the main theorem – these address actions of Alt ⁡ ( n ) \operatorname{Alt}(n) on 𝐿-groups of finite Morley rank, generically 2-transitive actions with abelian point stabilizers, and simple groups of rank 6.

Let 𝐺 be the finite simple group of Lie type G = E 7 ⁢ ( q ) G=E_{7}(q) , where 𝑞 is an odd prime power. Then 𝐺 is an index 2 subgroup of the adjoint group G ad G_{\operatorname{ad}} , which is also denoted by G ad = Inndiag ⁡ ( G ) G_{\operatorname{ad}}=\operatorname{Inndiag}(G) and known as the group of inner-diagonal automorphisms. It was proven by Cohen–Liebeck–Saxl–Seitz (1992) that there is an elementary abelian 2-subgroup 𝐸 of order 4 in G ad G_{\operatorname{ad}} such that N G ad ⁢ ( E ) / C G ad ⁢ ( E ) ≅ Sym 3 and C G ad ⁢ ( E ) = E × Inndiag ⁡ ( D 4 ⁢ ( q ) ) . N_{G_{\operatorname{ad}}}(E)/C_{G_{\operatorname{ad}}}(E)\cong\mathrm{Sym}_{3}\quad\text{and}\quad C_{G_{\operatorname{ad}}}(E)=E\times\operatorname{Inndiag}(D_{4}(q)). Furthermore, such an 𝐸 is unique up to conjugacy in G ad G_{\operatorname{ad}} . It is known that N G ⁢ ( E ) N_{G}(E) is always a maximal subgroup of 𝐺, and N G ad ⁢ ( E ) N_{G_{\operatorname{ad}}}(E) is a maximal subgroup of G ad G_{\operatorname{ad}} unless N G ad ⁢ ( E ) ≤ G N_{G_{\operatorname{ad}}}(E)\leq G . In this note, we describe the structure of N G ⁢ ( E ) N_{G}(E) . It turns out that we have N G ⁢ ( E ) = N G ad ⁢ ( E ) N_{G}(E)=N_{G_{\operatorname{ad}}}(E) if and only if q ≡ ± 1 mod 8 q\equiv\pm 1\bmod 8 .

Let 𝐺 be a finite solvable permutation group. Then, modulo a possibly trivial normal elementary abelian 3-subgroup, some set-stabilizer in 𝐺 is a 2-group.

For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.

In finite group theory, 𝑝-length is an important invariant for a 𝑝-solvable group. The famous Hall–Higman 𝑝-length theorem states that the 𝑝-length of a 𝑝-solvable group is bounded by the nilpotent class of its Sylow 𝑝-subgroups. In this paper, we improve this result by giving a better estimate on the 𝑝-length of a 𝑝-solvable group in terms of the 𝔘-class of its Sylow 𝑝-subgroups.

We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group Aut ⁡ ( G ) \operatorname{Aut}(G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order 2 9 2^{9} . All of them are Suzuki 2-groups, and they appear (in a different context) in an earlier classification of Dornhoff.

We present a structural description of finite nilpotent groups of class at most 2 using a specified number of subdirect and central products of such groups that are 2-generated. As a corollary, we show that any such group is isomorphic to a subgroup of a Heisenberg group satisfying certain properties. The motivation for these results is of a topological nature as they can be used to give lower bounds to the nilpotently Jordan property of the birational automorphism group of varieties and the homeomorphism group of compact manifolds.

In a 2018 paper, Behravesh and Delfani computed the minimal degree of a faithful permutation representation for groups of order p 5 p^{5} , where 𝑝 is an odd prime. However, their results are incorrect and incomplete: some groups are not considered, some minimal degrees are incorrectly determined, and the reported list of degrees is incomplete. We determine the minimal degree of a faithful permutation representation for each group of order p 5 p^{5} , where 𝑝 is an odd prime. We also record how to obtain such a representation.

Demonet, Iyama and Jasso introduced a new class of finite-dimensional algebras, 𝜏-tilting finite algebras . It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always 𝜏-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control 𝜏-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that 𝜏-tilting finiteness of a group algebra of a finite group 𝐺 is controlled by a 𝑝-hyperfocal subgroup of 𝐺 under some assumptions on 𝐺. We consider a group algebra of a finite group P ⋊ H P\rtimes H over an algebraically closed field of positive characteristic 𝑝, where 𝑃 is an abelian 𝑝-group and 𝐻 is an abelian p ′ p^{\prime} -group acting on 𝑃, and show that 𝑝-hyperfocal subgroups determine 𝜏-tilting finiteness of the group algebras in this case.

The goal of this paper is to show that, for any k ≥ 2 k\geq 2 , the subgroup of the automorphism group of a free group consisting of all automorphisms that act trivially on the ( k + 1 ) (k+1) -th derived quotient of the free group is not finitely generated in some stable range. In order to show this, we consider a slight generalization of results of Grossman in [Representations of the automorphism groups of free groups, J. Algebra 30 (1974), 388–399], and a sequence of descending filtrations of the IA-automorphism group of a free group.

In this paper, we prove that the outer automorphism groups of nonabelian topologically finitely generated free profinite groups are strongly indecomposable. This means that no open subgroup has a nontrivial direct product decomposition.

We show that the full mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for pure mapping class groups by Aramayona and Martínez Pérez. As a consequence of our result, we also obtain the proper geometric dimension of full spherical braid groups.

We study the connection between the fundamental groups of complex hyperbolic manifolds and those of spaces arising from the (relative) strict hyperbolization process due to Charney–Davis and Davis–Januszkiewicz–Weinberger. Viewing a non-uniform lattice Γ in PU ⁢ ( n , 1 ) \text{PU}(n,1) as a relatively hyperbolic group with respect to its cusp subgroups in the usual way, we show that, when n ≥ 2 n\geq 2 , Γ is not isomorphic to any relatively hyperbolic group arising from the relative strict hyperbolization process, via work of Lafont–Ruffoni. We also prove that a uniform lattice in PU ⁢ ( n , 1 ) \text{PU}(n,1) is not the fundamental group of a Charney–Davis strict hyperbolization when n ≥ 2 n\geq 2 , assuming the initial complex satisfies some mild conditions.