Volume 26, Issue 3

We study a certain family of simple fusion systems over finite 3-groups, ones that involve Todd modules of the Mathieu groups 2 ⁢ M 12 2M_{12} , M 11 M_{11} , and A 6 = O 2 ⁢ ( M 10 ) A_{6}=O^{2}(M_{10}) over F 3 \mathbb{F}_{3} , and show that they are all isomorphic to the 3-fusion systems of almost simple groups. As one consequence, we give new 3-local characterizations of Conway’s sporadic simple groups.

We will present an alternative approach to Zalesskiĭ’s theorem on diagonal embeddings of finite alternating groups.

We consider a finite, connected and simple graph Γ that admits a vertex-transitive group of automorphisms 𝐺. Under the assumption that, for all x ∈ V ⁢ ( Γ ) x\in V(\Gamma) , the local action G x Γ ⁢ ( x ) G_{x}^{\Gamma(x)} is the action of Sym ⁡ ( n ) \operatorname{Sym}(n) on ordered pairs, we show that the group G x [ 3 ] G_{x}^{[3]} , the point-wise stabiliser of a ball of radius three around 𝑥, is trivial.

In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.

A skew morphism of a finite group 𝐴 is a permutation 𝜑 of 𝐴 fixing the identity element and for which there is an integer-valued function 𝜋 on 𝐴 such that φ ⁢ ( x ⁢ y ) = φ ⁢ ( x ) ⁢ φ π ⁢ ( x ) ⁢ ( y ) \varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y) for all x , y ∈ A x,y\in A . In this paper, we restrict ourselves to the case when A = D n A=D_{n} , the dihedral group of order 2 ⁢ n 2n . Wang et al. [Smooth skew morphisms of dihedral groups, Ars Math. Contemp. 16 (2019), 2, 527–547] determined all 𝜑 under the condition that π ( φ ( x ) ) ≡ π ( x ) ( mod | φ | ) ) \pi(\varphi(x))\equiv\pi(x)\pmod{\lvert\varphi\rvert}) holds for every x ∈ D n x\in D_{n} , and later Kovács and Kwon [Regular Cayley maps for dihedral groups, J. Combin. Theory Ser. B 148 (2021), 84–124] characterised those 𝜑 such that there exists an inverse-closed ⟨ φ ⟩ \langle\varphi\rangle -orbit, which generates D n D_{n} . We show that these two types of skew morphisms comprise all skew morphisms of D n D_{n} . The result is used to classify the finite groups with a complementary factorisation into a dihedral and a core-free cyclic subgroup. As another application, a formula for the total number of skew morphisms of D p t D_{p^{t}} is also derived for any prime 𝑝.

It is a theorem of W. W. Comfort and K. A. Ross that if 𝐺 is a subgroup of a compact Abelian group and 𝑆 denotes the continuous homomorphisms from 𝐺 to the one-dimensional torus, then the topology on 𝐺 is the initial topology given by 𝑆. Assume that 𝐻 is a subgroup of 𝐺. We study how the choice of 𝑆 affects the topological placement and properties of 𝐻 in 𝐺. Among other results, we have made significant progress toward the solution of the following specific questions. How many totally bounded group topologies does 𝐺 admit such that 𝐻 is a closed (dense) subgroup? If C S C_{S} denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does C S C_{S} have a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC group ( topologically simple , resp.) if all its subgroups are closed (if 𝐺 and { e } \{e\} are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group 𝐺 admit?

Exploring inequalities involving the rank and relation gradients of pro-𝑝 modules and building upon recent results of Y. Antolín, A. Jaikin-Zapiran and M. Shusterman, we prove that every retract of a Demushkin group is inert in the sense of the Dicks–Ventura Inertia Conjecture.

For 0 < α ≤ 1 0<\alpha\leq 1 , we say that a sequence ( X k ) k > 0 (X_{k})_{k>0} of 𝑑-regular connected graphs has property D α D_{\alpha} if there exists a constant C > 0 C>0 such that diam ⁡ ( X k ) ≥ C ⋅ | X k | α \operatorname{diam}(X_{k})\geq C\cdot\lvert X_{k}\rvert^{\alpha} . We investigate property D α D_{\alpha} for arithmetic box spaces of the solvable Baumslag–Solitar groups BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) (with m ≥ 2 m\geq 2 ): these are box spaces obtained by embedding BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) into the upper triangular matrices in GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) and intersecting with a family M N k M_{N_{k}} of congruence subgroups of GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) , where the levels N k N_{k} are coprime with 𝑚 and N k ∣ N k + 1 N_{k}\mid N_{k+1} . We prove that if an arithmetic box space has D α D_{\alpha} , then α ≤ 1 2 \alpha\leq\frac{1}{2} ; if the family ( N k ) k (N_{k})_{k} of levels is supported on finitely many primes, the corresponding arithmetic box space has D 1 / 2 D_{\smash{1/2}} ; if the family ( N k ) k (N_{k})_{k} of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have D α D_{\alpha} for every α > 0 \alpha>0 . Moreover, we prove that if we embed BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) in the group of invertible upper-triangular matrices T n ⁢ ( Z ⁢ [ 1 / m ] ) T_{n}(\mathbb{Z}[1/m]) , then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).