Article Frontmatter was published on July 1, 2021 in the journal Journal of Group Theory (volume 24, issue 4).

Let 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄{\mathfrak{A}} such that the automorphism group of 𝔄{\mathfrak{A}} has at most c⁢nd⁢n{cn^{dn}} orbits in its componentwise action on the set of n -tuples with pairwise distinct entries, for some constants c,d{c,d} with d<1{d<1}. We show that 𝒦exp+{\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite

In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free

A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U⁢V\mathcal{UV}, where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application

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

In this paper, we introduce the notion of a quasi-powerful p -group for odd primes p . These are the finite p -groups G such that G/Z⁢(G){G/Z(G)} is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares many of the same properties as powerful p -groups. For example, we show that they have a regular power structure, and we generalise a result of Fernández-Alcober

Let G be a finite group, and let 𝔉{\mathfrak{F}} be a hereditary saturated formation. We denote by 𝐙𝔉⁢(G){\mathbf{Z}_{\mathfrak{F}}(G)} the product of all normal subgroups N of G such that every chief factor H/K{H/K} of G below N is 𝔉{\mathfrak{F}}-central in G , that is, (H/K)⋊(G/𝐂G⁢(H/K))∈𝔉{(H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F}}. A subgroup A⩽G{A\leqslant G} is said to be 𝔉{\

We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H∩KH\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G′′G^{\prime\prime} is finite.

Let G be isomorphic to GLn⁢(q){\mathrm{GL}_{n}(q)}, SLn⁢(q){\mathrm{SL}_{n}(q)}, PGLn⁢(q){\mathrm{PGL}_{n}(q)} or PSLn⁢(q){\mathrm{PSL}_{n}(q)}, where q=2a{q=2^{a}}. 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

Article Frontmatter was published on May 1, 2021 in the journal Journal of Group Theory (volume 24, issue 3).

We initiate the study of the groups (l,m∣n,k∣p,q)(l,m\mid n,k\mid p,q) defined by the presentation ⟨a,b∣al,bm,(a⁢b)n,(ap⁢bq)k⟩\langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle. When p=1p=1 and q=m-1q=m-1, we obtain the group (l,m∣n,k)(l,m\mid n,k), first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l=2l=2 and 1n+1k≤12\frac{1}{n}+\frac{1}{k}\leq\frac{1}{2}

Two subgroups A and B of a group G are said to be 𝒩{\mathcal{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{G=AB} and A and B are 𝒩{\mathcal{N}}-connected subgroups satisfying Max or Min.

In this paper, we determine the TI subgroups of the simple Suzuki groups Sz⁢(q)\mathrm{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

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{\frac{2}{3}n}. The proof depends on the classification of

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

Article Frontmatter was published on March 1, 2021 in the journal Journal of Group Theory (volume 24, issue 2).

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup Γ⁢(p){\Gamma(p)}, p a prime, is acted on by SL2⁢(𝔽p){\mathrm{SL}_{2}(\mathbb{F}_{p})}. Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by SL2⁢(𝔽p){\m

Let G be a finite group. An irreducible character of G is called a 𝒫{\mathcal{P}}-character if it is an irreducible constituent of (1H)G{(1_{H})^{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{\mathcal{P}}-characters.

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{x\in G} such that H⁢Tx=Tx⁢H{HT^{x}=T^{x}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〉{\langle H,T\rangle}. In this paper, we prove that

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.

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

Abstract We prove that there are no irreducible representations of B n {B_{n}} of dimension n + 1 {n+1} for n ⩾ 10 {n\geqslant 10} .

Abstract 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 ⋯ < X - n < ⋯ < X - 1 < X 0 < X 1 < ⋯ < X n < ⋯ \cdots

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

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

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ 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 $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq

Given a finite nonabelian semisimple group $G$, we describe those groups that have the same holomorph as $G$, that is, those regular subgroups $N\simeq G$ of $S(G)$, the group of permutations on the set $G$, such that $N_{S(G)}(N)=N_{S(G)}(\rho(G))$, where $\rho$ is the right regular representation of $G$.

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

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 $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts

Let $n$ be an integer greater than or equal to $3$ and $(R,\Delta)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,\Delta)$ normalised by a relative elementary subgroup $EU_{2n+1}((R,\Delta),(I,\Omega))$, then there is an odd form ideal $(J,\Sigma)$ such that $EU_{2n+1}((R,\Delta),(JI^{k},\Omega_{\min}^{JI^k}\over

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.

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian

Abstract 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 {\varGamma} be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of Γ {\varGamma} . In the case where G acts transitively on

Abstract 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 ) {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(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

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

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 Leder (arXiv:1810.06287) for finite cyclic groups, as well as resolving

The paper is concerned with the character theory of finite groups of Lie type. The set of irreducible characters of a group $G$ of Lie type is partitioned into the Lusztig series. We determine the maximum of the sizes of such series for classical groups with connected center.

We construct a few supercharacter theories for finite semidirect products with the normal subgroup of algebra group type. In the case of algebra groups, these supercharacter theories coincide with the one of P.Diaconis and I.M.Isaaks. For the parabolic subgroups of $GL(n)$, 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, but the depth function of which grows faster than any recursive function.

Abstract Assume G is a finite p-group. We prove that if all 𝒜 2 {\mathcal{A}_{2}} -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 𝒜 1 {\mathcal{A}_{1}} -subgroups and in which the intersection of every pair of distinct 𝒜 1 {\mathcal{A}_{1}} -subgroups equals the intersection of all the

Abstract 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 {p=2} by considering linear characters and those of even degree.

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

Abstract Let x be an element of a finite group G. It is clear that 〈 x 〉 ≤ C G ⁢ ( x ) ≤ G {\langle x\rangle\leq C_{G}(x)\leq G} . For the cases where C G ⁢ ( x ) = G {C_{G}(x)=G} and C G ⁢ ( x ) = 〈 x 〉 {C_{G}(x)=\langle x\rangle} for any element x of G, the structure of G is easily decided. In this paper, we investigate the case where C G ⁢ ( x ) {C_{G}(x)} is maximal in G and obtain the structure

Abstract We give a new purely algebraic approach to odd unitary groups using odd form rings. Using these objects, we give a self-contained proof of injective stability for the odd unitary K 1 {K_{1}} -functor under the stable rank condition.

Abstract Let G be a finite p-group, and let ⊗ 3 G {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of ⊗ 3 G {\otimes^{3}G} , 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

Abstract A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that G = A ⁢ B {G=AB} 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 {G=AB} with semisubnormal supersoluble subgroups

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

Abstract Let K = x G {K=x^{G}} 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 { | K n | } n ≥ 1 {\{\lvert K^{n}\rvert\}_{n\geq 1}} and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if | K 2 | < 3 2 ⁢ | K | {\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert} or if | K 4

Abstract Let 𝔽 {\mathbb{F}} be a finite field. We prove that the cohomology algebra H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} with coefficients in 𝔽 {\mathbb{F}} of a right-angled Artin group G Γ {G_{\Gamma}} is a strongly Koszul algebra for every finite graph Γ. Moreover, H ∙ ⁢ ( G Γ , 𝔽 ) {H^{\bullet}(G_{\Gamma},\mathbb{F})} is a universally Koszul algebra if, and only if, the graph

Abstract 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.

Abstract 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.

Abstract 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

Abstract 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

Abstract 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

Abstract 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

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G} . In this paper, we determine the finite simple groups of type E 6 ⁢ ( q ) {E_{6}(q)} and E 6 2 ⁢ ( q ) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional

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

Abstract If a group G is π-separable, where π is a set of primes, the set of irreducible characters B π ⁡ ( G ) ∪ B π ′ ⁡ ( G ) {\operatorname{B}_{\pi}(G)\cup\operatorname{B}_{\pi^{\prime}}(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

Abstract 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.

Abstract 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