Band 21, Heft 3

Let G be a simple algebraic group over an algebraically closed field K of characteristic p>0{p>0}. We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G . A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G . We generalize their result and show that if u has order p , then except for two known examples which occur in the case (G,p)=(C2,2){(G,p)=(C_{2},2)}, the subgroup X cannot be contained in a proper parabolic subgroup of G . In the case where u has order >p{>p}, we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group G are in one-to-one correspondence with the orbits of the action of Aut⁢(G){{\mathrm{Aut}}(G)} on the ordered generating pairs of G . If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group G and it is called uniquely regular. In this paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic p -groups G , i.e., the order of the automorphism group of G attains Hall’s upper bound. Maximally automorphic p -groups of nilpotency class three are classified.

We consider Majorana algebras generated by three Majorana axes a0{a_{0}}, a1{a_{1}} and a2{a_{2}} such that a0{a_{0}} and a1{a_{1}} generate a dihedral algebra of type 2⁢A{2A}. We show that such an algebra must occur as a Majorana representation of one of 27 groups. These 27 groups coincide with the subgroups of the Monster that are generated by three 2⁢A{2A}-involutions a , b and c such that ab is also a 2⁢A{2A}-involution, which were classified by S. P. Norton in 1985. Our work relies on that of S. Decelle and consists of showing that certain groups do not admit Majorana representations.

Let 𝐆{\mathbf{G}} be a connected reductive algebraic group over 𝔽¯p{\overline{\mathbb{F}}_{p}} and let F:𝐆→𝐆{F:\mathbf{G}\to\mathbf{G}} be a Frobenius endomorphism endowing 𝐆{\mathbf{G}} with an 𝔽q{\mathbb{F}_{q}}-rational structure. Bonnafé–Michel have shown that the Mackey formula for Deligne–Lusztig induction and restriction holds for the pair (𝐆,F){(\mathbf{G},F)} except in the case where q=2{q=2} and 𝐆{\mathbf{G}} has a quasi-simple component of type 𝖤6{\mathsf{E}_{6}}, 𝖤7{\mathsf{E}_{7}}, or 𝖤8{\mathsf{E}_{8}}. Using their techniques, we show that if q=2{q=2} and Z⁢(𝐆){Z(\mathbf{G})} is connected then the Mackey formula holds unless 𝐆{\mathbf{G}} has a quasi-simple component of type 𝖤8{\mathsf{E}_{8}}. This establishes the Mackey formula, for instance, in the case where (𝐆,F){(\mathbf{G},F)} is of type 𝖤7⁢(2){\mathsf{E}_{7}(2)}. Using this, together with work of Bonnafé–Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if Z⁢(𝐆){Z(\mathbf{G})} is connected.

The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G . The least element of the Chermak–Delgado lattice of G is known as the Chermak–Delgado subgroup of G . This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p -group lattices play a major role in the author’s constructions.

Let 𝔛{\mathfrak{X}} be a class of groups. A subgroup U of a group G is called 𝔛{\mathfrak{X}}- maximal in G provided that (a) U∈𝔛{U\in\mathfrak{X}}, and (b) if U≤V≤G{U\leq V\leq G} and V∈𝔛{V\in\mathfrak{X}}, then U=V{U=V}. A chief factor H/K{H/K} of G is called 𝔛{\mathfrak{X}}-eccentric in G provided (H/K)⋊G/CG⁢(H/K)∉𝔛{(H/K)\rtimes G/C_{G}(H/K)\not\in\mathfrak{X}}. A group G is called a quasi-𝔛{\mathfrak{X}}-group if for every 𝔛{\mathfrak{X}}-eccentric chief factor H/K{H/K} and every x∈G{x\in G}, x induces an inner automorphism on H/K{H/K}. We use 𝔛*{\mathfrak{X}^{*}} to denote the class of all quasi-𝔛{\mathfrak{X}}-groups. In this paper we describe all hereditary saturated formations 𝔉{\mathfrak{F}} containing all nilpotent groups such that the 𝔉*{\mathfrak{F}^{*}}-hypercenter of G coincides with the intersection of all 𝔉*{\mathfrak{F}^{*}}-maximal subgroups of G for every group G .

Let A be an elementary abelian r -group acting on a finite r′{r^{\prime}}-group G . Suppose that the fixed-point group CG⁡(a){\operatorname{C}_{G}(a)} is supersolvable for each a∈A#{a\in A^{\#}}. We show that G is supersolvable if |A|⩾r4{|A|\geqslant r^{4}} and that G′⩽𝐅3⁢(G){G^{\prime}\leqslant\mathbf{F}_{3}(G)} if |A|⩾r3{|A|\geqslant r^{3}}. Moreover, we prove some other results for cases when the fixed-point group CG⁡(a){\operatorname{C}_{G}(a)} is abelian, p -nilpotent or satisfies the Sylow tower property.

Let q be a prime, n a positive integer and A an elementary abelian group of order qr{q^{r}} with r≥2{r\geq 2} acting on a finite q′{q^{\prime}}-group G . We show that if all elements in γr-1⁢(CG⁢(a)){\gamma_{r-1}(C_{G}(a))} are n -Engel in G for any a∈A#{a\in A^{\#}}, then γr-1⁢(G){\gamma_{r-1}(G)} is k -Engel for some {n,q,r}{\{n,q,r\}}-bounded number k , and if, for some integer d such that 2d≤r-1{2^{d}\leq r-1}, all elements in the d th derived group of CG⁢(a){C_{G}(a)} are n -Engel in G for any a∈A#{a\in A^{\#}}, then the d th derived group G(d){G^{(d)}} is k -Engel for some {n,q,r}{\{n,q,r\}}-bounded number k . Assuming r≥3{r\geq 3}, we prove that if all elements in γr-2⁢(CG⁢(a)){\gamma_{r-2}(C_{G}(a))} are n -Engel in CG⁢(a){C_{G}(a)} for any a∈A#{a\in A^{\#}}, then γr-2⁢(G){\gamma_{r-2}(G)} is k -Engel for some {n,q,r}{\{n,q,r\}}-bounded number k , and if, for some integer d such that 2d≤r-2{2^{d}\leq r-2}, all elements in the d th derived group of CG⁢(a){C_{G}(a)} are n -Engel in CG⁢(a){C_{G}(a)} for any a∈A#,{a\in A^{\#},} then the d th derived group G(d){G^{(d)}} is k -Engel for some {n,q,r}{\{n,q,r\}}-bounded number k . Analogous (non-quantitative) results for profinite groups are also obtained.

We show that if π is the fundamental group of a 4-dimensional infrasolvmanifold then -2≤def⁢(π)≤0{-2\leq\mathrm{def}(\pi)\leq 0}, and give examples realizing each value allowed by our constraints, for each possible value of the rank of π/π′{\pi/\pi^{\prime}}. We also consider the abstract commensurators of such groups. Finally, we show that if G is a finitely generated group, the kernel of the natural homomorphism from G to its abstract commensurator Comm⁢(G){\mathrm{Comm}(G)} is locally nilpotent by locally finite, and is finite if def⁢(G)>1{\mathrm{def}(G)>1}.

Let A be an Artin group. A partition 𝒫{\mathcal{P}} of the set of standard generators of A is called admissible if, for all X,Y∈𝒫{X,Y\in\mathcal{P}}, X≠Y{X\neq Y}, there is at most one pair (s,t)∈X×Y{(s,t)\in X\times Y} which has a relation. An admissible partition 𝒫{\mathcal{P}} determines a quotient Coxeter graph Γ/𝒫{\Gamma/\mathcal{P}}. We prove that, if Γ/𝒫{\Gamma/\mathcal{P}} is either a forest or an even triangle free Coxeter graph and AX{A_{X}} is residually finite for all X∈𝒫{X\in\mathcal{P}}, then A is residually finite.