Volume 27, Issue 6

The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.

We say that a group 𝐺 is of profinite type if it can be realized as a Galois group of some field extension. Using Krull’s theory, this is equivalent to 𝐺 admitting a profinite topology. We also say that a group of profinite type is profinitely rigid if it admits a unique profinite topology. In this paper, we study when abelian groups and some group extensions are of profinite type or profinitely rigid. We also discuss the connection between the properties of profinite type and profinite rigidity to the injectivity and surjectivity of the cohomology comparison maps, which were studied by Sury and other authors.

Let 𝐺 be a group and 𝑘 a positive integer. We say that 𝐺 is conjugate separable abelian (CSA) if every maximal abelian subgroup of 𝐺 is malnormal. In this paper, as a natural generalization, we study groups with the property that all maximal nilpotent subgroups of class at most 𝑘 are malnormal, which we refer to as CSN 𝑘 groups, and we show that they have many properties in common with the more widely studied CSA groups. In addition, we introduce the class of nilpotency transitive groups of class 𝑘, denoted NT 𝑘 , and in the presence of a special residuality condition, we prove that the CSN 𝑘 and NT 𝑘 properties are equivalent.

Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H HX=XH for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT \mathrm{PT} -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT \mathrm{PT} is not inherited by subgroups, and 𝐺 is called a PT ̄ \overline{\mathrm{PT}} -group if all its subgroups have the PT \mathrm{PT} -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT \mathrm{PT} -property, then 𝐺 is a PT ̄ \overline{\mathrm{PT}} -group.

The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most 𝑛 involutions, then the group is finite and its order is bounded in terms of 𝑛 only. This answers a question of Strunkov from the Kourovka notebook.

Suppose that w = w ⁢ ( x 1 , … , x n ) w=w(x_{1},\ldots,x_{n}) is a word, i.e. an element of the free group F = ⟨ x 1 , … , x n ⟩ F=\langle x_{1},\ldots,x_{n}\rangle . The verbal subgroup w ⁢ ( G ) w(G) of a group 𝐺 is the subgroup generated by the set { w ⁢ ( x 1 , … , x n ) : x 1 , … , x n ∈ G } \{w(x_{1},\ldots,x_{n}):x_{1},\ldots,x_{n}\in G\} of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if | H : w ( H ) | < | G : w ( G ) | \lvert H:w(H)\rvert<\lvert G:w(G)\rvert for every H < G H<G . In this paper, we give new results on 𝑤-maximal groups, and study the weaker condition in which the previous inequality is not strict. Some applications are given: for example, if a finite group has a solvable (resp. nilpotent) section of size 𝑛, then it has a solvable (resp. nilpotent) subgroup of size at least 𝑛.

Let 𝐺 be a finite group, k ⁢ ( G ) k(G) the number of conjugacy classes of 𝐺, and 𝐵 a nilpotent subgroup of 𝐺. In this paper, we prove that | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) \lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G) if 𝐺 is solvable and that 15 7 ⁢ | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) \frac{15}{7}\lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G) if 𝐺 is nonsolvable, where π = π ⁢ ( B ) \pi=\pi(B) is the set of prime divisors of | B | \lvert B\rvert . Both bounds are best possible.

Let 𝐺 be a finite group, and let 𝐻 be a subgroup of 𝐺. We compute the probability, denoted by P G ⁢ ( H ) P_{G}(H) , that a left transversal of 𝐻 in 𝐺 is also a right transversal, thus a two-sided one. Moreover, we define, and denote by tp ⁡ ( G ) \operatorname{tp}(G) , the common transversal probability of 𝐺 to be the minimum, taken over all subgroups 𝐻 of 𝐺, of P G ⁢ ( H ) P_{G}(H) . We prove a number of results regarding the invariant tp ⁡ ( G ) \operatorname{tp}(G) , like lower and upper bounds, and possible values it can attain. We also show that tp ⁡ ( G ) \operatorname{tp}(G) determines structural properties of 𝐺. Finally, several open problems are formulated and discussed.

Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G 0 G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G 0 G_{0} in Ω, including the trivial orbit { α } \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

For k , ℓ ∈ N k,\ell\in\mathbb{N} , we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for k , ℓ ≥ 2 k,\ell\geq 2 . In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.

A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

Let 𝐺 be a finite simple group of Lie type and let 𝑇 be a maximal torus of 𝐺. It is well known that if the defining field of 𝐺 is large enough, then the normaliser of 𝑇 in 𝐺 is equal to the algebraic normaliser N ⁢ ( G , T ) N(G,T) . We identify explicitly all the cases when N G ⁢ ( T ) N_{G}(T) is not equal to N ⁢ ( G , T ) N(G,T) .