We study metabelian groups 𝐺 given by full rank finite presentations ⟨A∣R⟩M in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank max⁡{0,|A|-|R|} and a virtually abelian normal subgroup, and that if |R|≤|A|-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if |R|≥|A|. We further prove that if |R|≤|A|-1, then, in any direct

Two subgroups 𝐴 and 𝐵 of a group 𝐺 are said to be 𝒩-connected if, for all 𝑎 in 𝐴 and 𝑏 in 𝐵, the subgroup generated by 𝑎 and 𝑏 is a nilpotent group. In this paper, we study the structure of a group 𝐺 assuming that G=A⁢B and 𝐴 and 𝐵 are 𝒩-connected subgroups satisfying Max or Min.

The minimal degree of a permutation group 𝐺 is defined as the minimal number of non-fixed points of a non-trivial element of 𝐺. In this paper, we show that if 𝐺 is a transitive permutation group of degree 𝑛 having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of 𝐺 is at least 23⁢n. The proof depends on the classification of finite simple

Given a group 𝐺 of homeomorphisms of a first-countable Hausdorff space 𝒳, we prove that if the action of 𝐺 on 𝒳 is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in 𝒳 are conjugated by a homeomorphism of 𝒳. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal

We initiate the study of the groups (l,m∣n,k∣p,q) defined by the presentation ⟨a,b∣al,bm,(a⁢b)n,(ap⁢bq)k⟩. When p=1 and q=m-1, we obtain the group (l,m∣n,k), first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l=2 and 1n+1k≤12 and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions

Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation. We denote by ZF⁢(G) the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor H/K of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is, (H/K)⋊(G/CG⁢(H/K))∈F. A subgroup A⩽G is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain A=A0⩽A1⩽⋯⩽An=G such that either Ai-1⁢⊴⁢Ai

Let 𝐺 be isomorphic to GLn⁢(q), SLn⁢(q), PGLn⁢(q) or PSLn⁢(q), where q=2a. If 𝑡 is an involution lying in a 𝐺-conjugacy class 𝑋, then, for arbitrary 𝑛, we show that, as 𝑞 becomes large, the proportion of elements of 𝑋 which have odd order product with 𝑡 tends to 1. Furthermore, for 𝑛 at most 4, we give formulae for the number of elements in 𝑋 which have odd order product with 𝑡, in terms

Let 𝐺 be a finite group admitting an automorphism 𝜙 of prime order 𝑝 such that (|G|,p)=1. It is shown that if the fixed-point subgroup for 𝜙 has rank 𝑟 and (x-1⁢xϕ)e=1 for each x∈G, then the exponent of [G,ϕ] is (e,p,r)-bounded.

We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This process uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that

Given a finite nonabelian semisimple group 𝐺, we describe those groups that have the same holomorph as 𝐺, that is, those regular subgroups N≃G of S⁢(G), the group of permutations on the set 𝐺, such that NS⁢(G)⁢(N)=NS⁢(G)⁢(ρ⁢(G)), where 𝜌 is the right regular representation of 𝐺.

Journal Name: Journal of Group Theory Volume: 24 Issue: 1 Pages: i-iv

Let 𝐺 be a nonabelian group. We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets A1,A2,…,An of 𝐺 such that |Ai|⩾2 for each i=1,2,…,n. This paper investigates problems relating to groups with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a

We introduce two new notions of transitivity for Abelian 𝑝-groups based on isomorphism of quotients rather than the classical use of equality of height sequences associated with Abelian 𝑝-group theory. Unlike the classical theory where “most” groups are transitive, these new notions lead to much smaller classes, but even these classes are sufficiently large to be interesting.

Let 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1H)G for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

Let 𝑛 be an integer greater than or equal to 3, and let (R,Δ) be a Hermitian form ring, where 𝑅 is commutative. We prove that if 𝐻 is a subgroup of the odd-dimensional unitary group U2⁢n+1⁡(R,Δ) normalised by a relative elementary subgroup EU2⁢n+1⁡((R,Δ),(I,Ω)), then there is an odd form ideal (J,Σ) such that

We classify the locally compact second-countable (l.c.s.c.) groups 𝐴 that are abelian and topologically characteristically simple. All such groups 𝐴 occur as the monolith of some soluble l.c.s.c. group 𝐺 of derived length at most 3; with known exceptions (specifically, when 𝐴 is Qn or its dual for some n∈N), we can take 𝐺 to be compactly generated. This amounts to a classification of the possible

Let 𝐴 be a finite group acting by automorphisms on the finite group 𝐺. We introduce the commuting graph Γ⁢(G,A) of this action and study some questions related to the structure of 𝐺 under certain graph theoretical conditions on Γ⁢(G,A).

In this paper, we determine the TI subgroups of the simple Suzuki groups Sz⁢(q). More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively

A subgroup 𝐻 of a group 𝐺 is said to be conditionally permutable (or 𝑐-permutable for short) in 𝐺 if, for every subgroup 𝑇 of 𝐺, there exists an element x∈G such that H⁢Tx=Tx⁢H. A subgroup 𝐻 of a group 𝐺 is said to be completely 𝑐-permutable in 𝐺 if, for every subgroup 𝑇 of 𝐺, the subgroups 𝐻 and 𝑇 are 𝑐-permutable in ⟨H,T⟩. In this paper, we prove that H/HG is nilpotent if 𝐻 is a completely

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group 𝐺 of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group 𝐺 in terms of its Lie rank and defining characteristic. When 𝐺 is specified

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

We prove that the Fibonacci group F⁢(2,n) for 𝑛 odd and n≥11 is hyperbolic. We do this by applying a curvature argument to an arbitrary van Kampen diagram of F⁢(2,n) and show that it satisfies a linear isoperimetric inequality. It then follows that F⁢(2,n) is hyperbolic.

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products

Journal Name: Journal of Group Theory Volume: 23 Issue: 6 Pages: i-iv

We construct a few supercharacter theories for finite semidirect products where the normal subgroup is of algebra group type. In the case of algebra groups, these supercharacter theories coincide with those of P. Diaconis and I. M. Isaacs. For the parabolic subgroups of GL⁢(n,Fq), the supercharacters and superclasses are classified.

We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which

Assume G is a finite p-group. We prove that if all 𝒜2-subgroups of G are generated by two elements, then so are all non-abelian subgroups of G. By using this result, we classify the p-groups which have at least two 𝒜1-subgroups and in which the intersection of every pair of distinct 𝒜1-subgroups equals the intersection of all the 𝒜1-subgroups. It turns out that such p-groups are the finite p-groups

The Ito–Michler theorem on character degrees states that if a prime p does not divide the degree of any irreducible character of a finite group G, then G has a normal Sylow p-subgroup. We give some strengthened versions of this result for p=2 by considering linear characters and those of even degree.

Journal Name: Journal of Group Theory Volume: 23 Issue: 5 Pages: i-iv

We prove that there are no irreducible representations of Bn of dimension n+1 for n⩾10.

Let 𝔽 be a finite field. We prove that the cohomology algebra H∙⁢(GΓ,𝔽) with coefficients in 𝔽 of a right-angled Artin group GΓ is a strongly Koszul algebra for every finite graph Γ. Moreover, H∙⁢(GΓ,𝔽) is a universally Koszul algebra if, and only if, the graph Γ associated to the group GΓ has the diagonal property. From this, we obtain several new examples of pro-p groups, for a prime number p

If θ is a subgroup property, a group G is said to satisfy the double chain condition on θ-subgroups if it admits no infinite double sequences

M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput.42 (2007), 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

We generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra 163 1994, 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set B in the positive system

Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups. A group is invariably generated by a subset 𝑆 if replacing each

A protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group is equal to the growth rate of the induced endomorphism on its abelianization, generalizing the corresponding result for an automorphism in [T. Koberda, Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups, In the Tradition of Ahlfors–Bers. VI, Contemp. Math. 590, American

A subgroup H of a group G is said to be pronormal in G if H and Hg are conjugate in 〈H,Hg〉 for every g∈G. In this paper, we determine the finite simple groups of type E6⁢(q) and E62⁢(q) in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let Γ be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of Γ. In the case where G acts transitively on the edge set and quasiprimitively

The augmentation powers in an integral group ring ℤ⁢G induce a natural filtration of the unit group of ℤ⁢G analogous to the filtration of the group G given by its dimension series {Dn⁢(G)}n≥1. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

If a group G is π-separable, where π is a set of primes, the set of irreducible characters Bπ⁡(G)∪Bπ′⁡(G) can be defined. In this paper, we prove variants of some classical theorems in character theory, namely the theorem of Ito–Michler and Thompson’s theorem on character degrees, involving irreducible characters in the set Bπ⁡(G)∪Bπ′⁡(G).

In this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.

Given a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the

Journal Name: Journal of Group Theory Volume: 23 Issue: 4 Pages: i-iv

Let G be a p-group, and let χ be an irreducible character of G. The codegree of χ is given by |G:ker(χ)|/χ(1). Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2. Here we investigate p-groups with exactly four codegrees. If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree

Let cs⁢(G) denote the set of conjugacy class sizes of a group G, and let cs*⁢(G)=cs⁢(G)∖{1} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) cs⁢(G)={a,a+d,…,a+r⁢d} is an arithmetic progression with r⩾2; (2) cs*⁢(G)={2,4,6} is the smallest case where cs*⁢(G) is an arithmetic progression of length more than 2 (our most substantial result); (3) the

A semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup. A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous. If S is a nondiscrete monothetic semitopological semigroup, then the set S′ of all limit points of S is a closed ideal of S. Let S be a locally compact

In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups

For a solvable group, a theorem of Gaschutz shows that F⁢(G)/Φ⁢(G) is a direct sum of irreducible G-modules and a faithful G/F⁢(G)-module. If each of these irreducible modules is primitive, we show that every non-vanishing element of G lies in F⁢(G).

Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation xn=1 has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences

The function FG⁢(n) gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for FN⁢(n)

Let K=xG be the conjugacy class of an element x of a group G, and suppose K is finite. We study the increasing sequence of natural numbers {|Kn|}n≥1 and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if |K2|<32⁢|K| or if |K4|<2⁢|K|, then Kn is a coset of the normal subgroup [x,G] for all n≥2 or 4, respectively. We then use these results to contribute

Additivity with respect to exact sequences is, notoriously, a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by using the structure theorems for such groups in an essential way. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity

Let G be a finite p-group, and let ⊗3G be its triple tensor product. In this paper, we obtain an upper bound for the order of ⊗3G, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained

A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that G=A⁢B and AX is a subgroup of G for every subgroup X of B. We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semisubnormal in G if A is subnormal in G or seminormal in G. A group G=A⁢B with semisubnormal supersoluble subgroups A and B is studied. The

Based on computing evidence, Guarnieri and Vendramin conjectured that, for a generalized quaternion group G of order 2n⩾32, there are exactly seven isomorphism classes of braces with adjoint group G. The conjecture is proved in the paper.

Let G be a finite transitive permutation group of degree n, with point stabilizer H≠1 and permutation character π. For every positive integer t, we consider the generalized character ψt=ρG-t⁢(π-1G), where ρG is the regular character of G and 1G the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that ψt is a character of G. A necessary condition is that t≤min⁡{n-1

G. Navarro raised the question of when two vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup. The present note provides a sufficient criterion for this to happen. This generalises a result by Navarro for simple modules over finite p-solvable groups, which is the main motivation for this note.

Journal Name: Journal of Group Theory Volume: 23 Issue: 3 Pages: i-iv