Volume 22, Issue 4

We extend the results of our earlier work on Jordan-type bounds for finite subgroups of complex classical groups to real groups. The bounds that we obtain are related to our previous results by means of structural results for finite linear groups that we can generalise here to compact groups. For the real analogue of orthogonal groups, we take into account the signature of a real quadratic form to determine bounds in every case.

Robinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

In this paper, we prove that for any positive integers n,s,t{n,s,t} such that n≥10{n\geq 10}, s,t≥2{s,t\geq 2} and n-1≥s+t{n-1\geq s+t}, there exists a regular polytope with Schläfli type {2s,2t}{\{2^{s},2^{t}\}} and its automorphism group is of order 2n{2^{n}}. Furthermore, we classify regular polytopes with automorphism groups of order 2n{2^{n}} and Schläfli types {4,2n-3},{4,2n-4}{\{4,2^{n-3}\},\{4,2^{n-4}\}} and {4,2n-5}{\{4,2^{n-5}\}}, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hungar. 53 2006, 1–2, 231–255].

A skew-morphism of a finite group A is a permutation φ on A fixing the identity element, and for which there exists an integer function π on A such that, for all x,y∈A{x,y\in A}, φ⁢(x⁢y)=φ⁢(x)⁢φπ⁢(x)⁢(y){\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)}. In [I. Kovács and R. Nedela, Skew-morphisms of cyclic p -groups, J. Group Theory 20 2017, 6, 1135–1154], Kovács and Nedela determined skew-morphisms of the cyclic p -groups for any odd prime p . In this paper, we shall determine that of cyclic 2-groups.

Let G be a finite group. G is called a Frobenius–Wielandt group if there exists H<G{H<G} such that U=〈H∩Hg∣g∈G-H〉{U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H . The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space G/H{G/H}, it states in particular that the subgroup D=DG⁢(H){D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G . Let W=UG{W=U^{G}}, the normal closure of U in G . Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality W∩H=U{W\cap H=U}. This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with H<G{H<G}: (a) W=D{W=D} and G=H⁢W{G=HW}, or (b) W<D{W<D} and H⁢W<G{HW<G}. In the latter case, G/W{G/W} is a Frobenius group with a Frobenius complement H⁢W/W{HW/W} and Frobenius kernel D/W{D/W}.

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p -groups of order p5{p^{5}}. As an application, we determine the capability of groups of order p5{p^{5}}.

We determine all saturated fusion systems ℱ{\mathcal{F}} on a Sylow 3-subgroup of the sporadic McLaughlin group that do not contain any non-trivial normal 3-subgroup and show that they are all realizable.

In the paper, we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that, if G≃PSLnε⁡(q){G\simeq\operatorname{PSL}_{n}^{\varepsilon}(q)}, then, for every power word of type xM{x^{M}}, there exist constants c and N such that |ω⁢(G)|>c⁢ln⁡(n)⁢|G|n{\lvert\omega(G)\rvert>c\frac{\ln(n)\lvert G\rvert}{n}} whenever |G|>N{\lvert G\rvert>N}.

We describe the structure of locally finite groups of finite centralizer dimension.

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.

A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let (X,ℰ){(X,\mathcal{E})} be a coarse space, and let 𝔐{\mathfrak{M}} be a variety of groups different from the variety of singletons. We prove that there is a coarse group F𝔐⁢(X,ℰ)∈𝔐{F_{\mathfrak{M}}(X,\mathcal{E})\in\mathfrak{M}} such that (X,ℰ){(X,\mathcal{E})} is a subspace of F𝔐⁢(X,ℰ){F_{\mathfrak{M}}(X,\mathcal{E})}, X generates F𝔐⁢(X,ℰ){F_{\mathfrak{M}}(X,\mathcal{E})} and every coarse mapping (X,ℰ)→(G,ℰ′){(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}, where G∈𝔐{G\in\mathfrak{M}}, (G,ℰ′){(G,\mathcal{E}^{\prime})} is a coarse group, can be extended to coarse homomorphism F𝔐⁢(X,ℰ)→(G,ℰ′){F_{\mathfrak{M}}(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}. If 𝔐{\mathfrak{M}} is the variety of all groups, the groups F𝔐⁢(X,ℰ){F_{\mathfrak{M}}(X,\mathcal{E})} are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.