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Cliques in derangement graphs for innately transitive groups J. Group Theory (IF 0.5) Pub Date : 2024-03-14 Marco Fusari, Andrea Previtali, Pablo Spiga
Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function f : N → N f\colon\mathbb{N}\to\mathbb{N}
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A generalization of the Brauer–Fowler theorem J. Group Theory (IF 0.5) Pub Date : 2024-03-14 Saveliy V. Skresanov
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
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Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups J. Group Theory (IF 0.5) Pub Date : 2024-03-08 Owen Garnier
We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type
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Representation zeta function of a family of maximal class groups: Non-exceptional primes J. Group Theory (IF 0.5) Pub Date : 2024-03-07 Shannon Ezzat
We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups M n M_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p < n p
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Character degrees of 5-groups of maximal class J. Group Theory (IF 0.5) Pub Date : 2024-03-05 Lijuan He, Heng Lv, Dongfang Yang
Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either
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Automorphic word maps and the Amit–Ashurst conjecture J. Group Theory (IF 0.5) Pub Date : 2024-02-15 Harish Kishnani, Amit Kulshrestha
In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class
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The central tree property and algorithmic problems on subgroups of free groups J. Group Theory (IF 0.5) Pub Date : 2024-02-13 Mallika Roy, Enric Ventura, Pascal Weil
We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic
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The relational complexity of linear groups acting on subspaces J. Group Theory (IF 0.5) Pub Date : 2024-02-13 Saul D. Freedman, Veronica Kelsey, Colva M. Roney-Dougal
The relational complexity of a subgroup 𝐺 of Sym ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{k} for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL n ( F ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ( F
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Finite normal subgroups of strongly verbally closed groups J. Group Theory (IF 0.5) Pub Date : 2024-02-13 Filipp D. Denissov
In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally
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Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups J. Group Theory (IF 0.5) Pub Date : 2024-02-09 Victor S. Monakhov, Irina L. Sokhor
A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then G / F ( G ) G/F(G) is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.
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Realizing finite groups as automizers J. Group Theory (IF 0.5) Pub Date : 2024-02-08 Sylvia Bayard, Justin Lynd
It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.
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The algebraic entropy of one-dimensional finitary linear cellular automata J. Group Theory (IF 0.5) Pub Date : 2024-01-30 Hasan Akın, Dikran Dikranjan, Anna Giordano Bruno, Daniele Toller
The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on Z m \mathbb{Z}_{m} from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual S ̂ \hat{S} of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on Z m \mathbb{Z}_{m} ; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear
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Units, zero-divisors and idempotents in rings graded by torsion-free groups J. Group Theory (IF 0.5) Pub Date : 2024-01-24 Johan Öinert
The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons
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Multiple transitivity except for a system of imprimitivity J. Group Theory (IF 0.5) Pub Date : 2024-01-18 Colin D. Reid
Let Ω be a set equipped with an equivalence relation ∼ \sim ; we refer to the equivalence classes as blocks of Ω. A permutation group G ≤ Sym ( Ω ) G\leq\mathrm{Sym}(\Omega) is 𝑘-by-block-transitive if ∼ \sim is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on
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Polynomial maps and polynomial sequences in groups J. Group Theory (IF 0.5) Pub Date : 2024-01-15 Ya-Qing Hu
This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete
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Sublinearly Morse boundary of CAT(0) admissible groups J. Group Theory (IF 0.5) Pub Date : 2024-01-15 Hoang Thanh Nguyen, Yulan Qing
We show that if 𝐺 is an admissible group acting geometrically on a CAT ( 0 ) \operatorname{CAT}(0) space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary ( ∂ κ G , ν ) (\partial_{\kappa}G,\nu) is a model for the Poisson boundary of ( G , μ ) (G,\mu) , where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.
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Action of automorphisms on irreducible characters of finite reductive groups of type 𝖠 J. Group Theory (IF 0.5) Pub Date : 2024-01-15 Farrokh Shirjian, Ali Iranmanesh, Farideh Shafiei
Let 𝐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 𝖠. In this paper, we give an explicit description of the action of automorphisms of 𝐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay
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Orders on free metabelian groups J. Group Theory (IF 0.5) Pub Date : 2023-11-29 Wenhao Wang
A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover,
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Virtual planar braid groups and permutations J. Group Theory (IF 0.5) Pub Date : 2023-11-29 Tushar Kanta Naik, Neha Nanda, Mahender Singh
Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander–Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids
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An exact sequence for the graded Picent J. Group Theory (IF 0.5) Pub Date : 2023-11-29 Andrei Marcus, Virgilius-Aurelian Minuță
To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picent gr ( A ) \mathrm{Picent}^{\mathrm{gr}}(A) of isomorphism classes of invertible 𝐺-graded ( A , A ) (A,A) -bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a Picent \mathrm{Picent} version of the Beattie–del Río exact sequence, involving Dade’s group G [ B ] G[B] , which relates Picent gr ( A )
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Invariance of the Schur multiplier, the Bogomolov multiplier and the minimal number of generators under a variant of isoclinism J. Group Theory (IF 0.5) Pub Date : 2023-11-27 Ammu E. Antony, Sathasivam Kalithasan, Viji Z. Thomas
We introduce the 𝑞-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under 𝑞-isoclinism. We prove that the 𝑞-Schur multiplier is invariant under 𝑞-exterior isoclinism, and as an easy consequence, we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if 𝐺 and 𝐻 are 𝑝-groups with G / Z ∧ ( G ) ≅ H /
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Finite groups with modular 𝜎-subnormal subgroups J. Group Theory (IF 0.5) Pub Date : 2023-11-17 A-Ming Liu, Mingzhu Chen, Inna N. Safonova, Alexander N. Skiba
Let 𝜎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 𝜎-subnormal subgroup is modular.
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Structure of the Macdonald groups in one parameter J. Group Theory (IF 0.5) Pub Date : 2023-11-07 Alexander Montoya Ocampo, Fernando Szechtman
Consider the Macdonald groups G ( α ) = ⟨ A , B ∣ A [ A , B ] = A α , B [ B , A ] = B α ⟩ G(\alpha)=\langle A,B\mid A^{[A,B]}=A^{\alpha},\,B^{[B,A]}=B^{\alpha}\rangle , α ∈ Z \alpha\in\mathbb{Z} . We fill a gap in Macdonald’s proof that G ( α ) G(\alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of G ( α ) G(\alpha)
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The Reidemeister spectrum of direct products of nilpotent groups J. Group Theory (IF 0.5) Pub Date : 2023-11-01 Pieter Senden
We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete
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Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups J. Group Theory (IF 0.5) Pub Date : 2023-10-18 Luis Paris, Olga Varghese
By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.
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Finitely generated metabelian groups arising from integer polynomials J. Group Theory (IF 0.5) Pub Date : 2023-09-26 Derek J. S. Robinson
It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.
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Hall classes in linear groups J. Group Theory (IF 0.5) Pub Date : 2023-09-20 Francesco de Giovanni, Marco Trombetti, Bertram A. F. Wehrfritz
A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that G / N ′ G/N^{\prime} is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that G / N ′ G/N^{\prime} belongs to 𝔛. Examples have been given in [F. de Giovanni, M. Trombetti and B. A.
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A closure operator on the subgroup lattice of GL(𝑛,𝑞) and PGL(𝑛,𝑞) in relation to the zeros of the Möbius function J. Group Theory (IF 0.5) Pub Date : 2023-09-18 Luca Di Gravina
Let F q \mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional F q \mathbb{F}_{q} -vector space V = F q n V=\mathbb{F}_{q}^{n} . In this paper, we define a closure operator on the subgroup lattice of the group G = PGL ( V ) G=\mathrm{PGL}(V) . Let 𝜇 denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups 𝐻 of 𝐺
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Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability J. Group Theory (IF 0.5) Pub Date : 2023-09-18 Naoko Kunugi, Kyoichi Suzuki
We discuss representations of finite groups having a common central 𝑝-subgroup 𝑍, where 𝑝 is a prime number. For the principal 𝑝-blocks, we give a method of constructing a relative 𝑍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann’s results on stable
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On the strong connectivity of the 2-Engel graphs of almost simple groups J. Group Theory (IF 0.5) Pub Date : 2023-09-18 Francesca Dalla Volta, Fabio Mastrogiacomo, Pablo Spiga
The Engel graph of a finite group 𝐺 is a directed graph encoding the pairs of elements in 𝐺 satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.
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Graphical complexes of groups J. Group Theory (IF 0.5) Pub Date : 2023-09-13 Tomasz Prytuła
We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear CAT ( 0 ) \mathrm{CAT}(0) , C ( 6 ) C(6) graphical small cancellation, and a systolic one. We then
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Finite 𝑝-groups of class two with a small multiple holomorph J. Group Theory (IF 0.5) Pub Date : 2023-09-05 A. Caranti, Cindy (Sin Yi) Tsang
We consider the quotient group T ( G ) T(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T ( G ) T(G) contains a cyclic subgroup of order p r − 1 ( p − 1 ) p^{r-1}(p-1) , where p r p^{r} is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with
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Presentations of Schur covers of braid groups J. Group Theory (IF 0.5) Pub Date : 2023-08-15 Toshiyuki Akita, Rikako Kawasaki, Takao Satoh
In this paper, we consider several basic facts of Schur covers of the symmetric groups and braid groups. In particular, we give explicit presentations of Schur covers of braid groups.
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A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees J. Group Theory (IF 0.5) Pub Date : 2023-08-04 Yong Yang
The codegree of a character 𝜒 of a finite group 𝐺 is cod ( χ ) := | G : ker ( χ ) | χ ( 1 ) . \operatorname{cod}(\chi):=\frac{\lvert G:\ker(\chi)\rvert}{\chi(1)}. We show that the set of codegrees of the Ree groups F 4 2 ( q 2 ) {}^{2}F_{4}(q^{2}) ( q 2 = 2 2 n + 1 q^{2}=2^{2n+1} , n ≥ 1 n\geq 1 ) determines the groups up to isomorphism.
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Elementary amenable groups of cohomological dimension 3 J. Group Theory (IF 0.5) Pub Date : 2023-08-02 Jonathan A. Hillman
We show that torsion-free elementary amenable groups of Hirsch length at most 3 are solvable, of derived length at most 3. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products BS ( 1 , n ) ⋊ Z \mathrm{BS}(1,n)\rtimes\mathbb{Z} , or properly ascending HNN extensions with base Z 2 \mathbb{Z}^{2}
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On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′ J. Group Theory (IF 0.5) Pub Date : 2023-07-28 Danila O. Revin, Andrei V. Zavarnitsine
We prove that if L = F 4 2 ( 2 2 n + 1 ) ′ L={}^{2}F_{4}(2^{2n+1})^{\prime} and 𝑥 is a nonidentity automorphism of 𝐿, then G = ⟨ L , x ⟩ G=\langle L,x\rangle has four elements conjugate to 𝑥 that generate 𝐺. This result is used to study the following conjecture about the 𝜋-radical of a finite group. Let 𝜋 be a proper subset of the set of all primes and let 𝑟 be the least prime not belonging
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Quantifying lawlessness in finitely generated groups J. Group Theory (IF 0.5) Pub Date : 2023-07-25 Henry Bradford
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N \mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N} . We show that A Γ \mathcal{A}_{\Gamma} is bounded if and only if Γ has a non-abelian free subgroup. By contrast, we construct, for any non-decreasing unbounded function f : N → N f\colon\mathbb{N}\to\mathbb{N} , an
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A characterization of finite groups having a single Galois conjugacy class of certain irreducible characters J. Group Theory (IF 0.5) Pub Date : 2023-07-24 Yu Zeng, Dongfang Yang
Let 𝐺 be a finite group and let Irr s ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) be the set of irreducible complex characters 𝜒 of 𝐺 such that χ ( 1 ) 2 \chi(1)^{2} does not divide the index of the kernel of 𝜒. In this paper, we classify the finite groups 𝐺 for which any two characters in Irr s ( G ) \mathrm{Irr}_{\mathfrak{s}}(G) are Galois conjugate. In particular, we show that such groups are
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A classification of the prime graphs of pseudo-solvable groups J. Group Theory (IF 0.5) Pub Date : 2023-07-18 Ziyu Huang, Thomas Michael Keller, Shane Kissinger, Wen Plotnick, Maya Roma, Yong Yang
The prime graph Γ ( G ) \Gamma(G) of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | \lvert G\rvert , and p - q p\textup{-}q is an edge in Γ ( G ) \Gamma(G) if and only if 𝐺 has an element of order p q pq . Since their inception in the 1970s, these graphs have been studied extensively; however, completely classifying the possible
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A note on 𝑑-maximal 𝑝-groups J. Group Theory (IF 0.5) Pub Date : 2023-07-04 Messab Aiech, Hanifa Zekraoui, Yassine Guerboussa
A finite 𝑝-group 𝐺 is said to be 𝑑-maximal if d ( H ) < d ( G ) d(H)
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On multidimensional Schur rings of finite groups J. Group Theory (IF 0.5) Pub Date : 2023-07-04 Gang Chen, Qing Ren, Ilia Ponomarenko
For any finite group 𝐺 and a positive integer 𝑚, we define and study a Schur ring over the direct power G m G^{m} , which gives an algebraic interpretation of the partition of G m G^{m} obtained by the 𝑚-dimensional Weisfeiler–Leman algorithm. It is proved that this ring determines the group 𝐺 up to isomorphism if m ≥ 3 m\geq 3 , and approaches the Schur ring associated with the group Aut ( G
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Class-two quotients of finite permutation groups J. Group Theory (IF 0.5) Pub Date : 2023-06-28 Hangyang Meng, Xiuyun Guo
Let 𝐺 be a permutation group on a finite set and let 𝑝 be a prime. In this paper, we prove that the largest class-two 𝑝-quotient of 𝐺 has order at most p n / p p^{n/p} (or 2 3 n / 4 2^{3n/4} if p = 2 p=2 ), where 𝑛 is the number of points moved by a Sylow 𝑝-subgroup of 𝐺. Further, we describe the groups whose largest class-two 𝑝-quotients can reach such a bound. This extends earlier work
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The nilpotent genus of finitely generated residually nilpotent groups J. Group Theory (IF 0.5) Pub Date : 2023-06-28 Niamh O’Sullivan
Let 𝐺 and 𝐻 be residually nilpotent groups. Then 𝐺 and 𝐻 are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A potentially stronger condition is that 𝐻 is para-𝐺 if there exists a monomorphism of 𝐺 into 𝐻 which induces isomorphisms between the corresponding quotients of their lower central series. We first consider finitely generated residually
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More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups J. Group Theory (IF 0.5) Pub Date : 2023-06-26 Wei-Juan Zhang
In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } \{4,4,4\} with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , … , 4 } \{4,4,\dots,4\} with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type.
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Projective representations of Heisenberg groups over the rings of order 𝑝2 J. Group Theory (IF 0.5) Pub Date : 2023-06-26 Sumana Hatui, E. K. Narayanan, Pooja Singla
We describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order p 2 p^{2} . We also describe all projective representations of Heisenberg groups with entries from the rings Z / p 2 Z \mathbb{Z}/p^{2}\mathbb{Z} and F p [ t ] / ( t 2 ) \mathbb{F}_{p}[t]/(t^{2}) for odd primes 𝑝 and obtain a classification of their degenerate and
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Commutator endomorphisms of totally projective abelian 𝑝-groups J. Group Theory (IF 0.5) Pub Date : 2023-06-22 Patrick Keef
For a primary abelian group 𝐺, Chekhlov and Danchev (2015) defined three variations of Kaplansky’s notion of full transitivity by restricting one’s attention to the subgroup, the subring and the unitary subring of the endomorphism ring of 𝐺 generated by the collection of all commutator endomorphisms. They posed the problem of describing exactly which totally projective groups exhibit these forms
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Redundant relators in cyclic presentations of groups J. Group Theory (IF 0.5) Pub Date : 2023-06-21 Ihechukwu Chinyere, Gerald Williams
A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and show that if
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On weak commutativity in 𝑝-groups J. Group Theory (IF 0.5) Pub Date : 2023-06-14 Raimundo Bastos, Emerson de Melo, Ricardo de Oliveira, Carmine Monetta
The weak commutativity group χ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We present new bounds for the exponent of χ ( G ) \chi(G) and its sections, when 𝐺 is a finite 𝑝-group.
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The Higman operations and embeddings of recursive groups J. Group Theory (IF 0.5) Pub Date : 2023-06-12 Vahagn H. Mikaelian
In the context of Higman embeddings of recursive groups into finitely presented groups, we suggest an approach, termed the 𝐻-machine, which for certain wide classes of groups allows constructive Higman embeddings of recursive groups into finitely presented groups. The approach is based on Higman operations, and it explicitly constructs some specific recursively enumerable sets of integer sequences
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5-Regular prime graphs of finite nonsolvable groups J. Group Theory (IF 0.5) Pub Date : 2023-06-07 Qinghong Guo, Weijun Liu, Lu Lu
The prime graph Δ ( G ) \Delta(G) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and 𝑞 are joined by an edge if the product p q pq divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31]
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On the converse of Gaschütz’ complement theorem J. Group Theory (IF 0.5) Pub Date : 2023-05-10 Benjamin Sambale
Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G N\leq H\leq G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 (\lvert N\rvert,\lvert G:H\rvert)=1 . If 𝑁 is abelian, a theorem of Gaschütz asserts that 𝑁 has a complement in 𝐺 as well. Brandis has asked whether the commutativity of 𝑁 can be replaced by some weaker property. We prove that 𝑁 has a complement in 𝐺
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Morse boundaries of graphs of groups with finite edge groups J. Group Theory (IF 0.5) Pub Date : 2023-05-02 Stefanie Zbinden
In this paper, we prove that the Morse boundary of a free product depends only on the Morse boundary of its factors. In fact, we also prove the analogous result for graphs of groups with finite edge groups and infinitely many ends. This generalizes earlier work of Martin and Świątkowski in the case of non-hyperbolic groups.
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The number of locally invariant orderings of a group J. Group Theory (IF 0.5) Pub Date : 2023-04-28 Idrissa Ba, Adam Clay, Ian Thompson
We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group, we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along
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Algebraic groups over finite fields: Connections between subgroups and isogenies J. Group Theory (IF 0.5) Pub Date : 2023-03-24 Davide Sclosa
Let 𝐺 be a linear algebraic group defined over a finite field F q \mathbb{F}_{q} . We present several connections between the isogenies of 𝐺 and the finite groups of rational points ( G ( F q n ) ) n ≥ 1 (G(\mathbb{F}_{\smash{q^{n}}}))_{n\geq 1} . We show that an isogeny ϕ : G ′ → G \phi\colon G^{\prime}\to G over F q \mathbb{F}_{q} gives rise to a subgroup of fixed index in G ( F q n ) G(\m
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On limits of betweenness relations J. Group Theory (IF 0.5) Pub Date : 2023-03-20 David Bradley-Williams, John K. Truss
We give a flexible method for constructing a wide variety of limits of betweenness relations. This unifies work of Adeleke, who constructed a Jordan group preserving a limit of betweenness relations, and Bhattacharjee and Macpherson who gave an alternative method using a Fraïssé-type construction. A key ingredient in their work is the notion of a tree of 𝐵-sets. We employ this and extend its use to
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Hochschild cohomology of symmetric groups and generating functions J. Group Theory (IF 0.5) Pub Date : 2023-03-08 David Benson, Radha Kessar, Markus Linckelmann
In this article, we compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups. We show that the generating function of the dimensions of the Hochschild cohomology in any fixed degree of
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Stiefel–Whitney classes of representations of SL(2, 𝑞) J. Group Theory (IF 0.5) Pub Date : 2023-02-27 Neha Malik, Steven Spallone
We describe the Stiefel–Whitney classes (SWCs) of orthogonal representations 𝜋 of the finite special linear groups G = SL ( 2 , F q ) G=\operatorname{SL}(2,\mathbb{F}_{q}) , in terms of character values of 𝜋. From this calculation, we can answer interesting questions about SWCs of 𝜋. For instance, we determine the subalgebra of H * ( G , Z / 2 Z ) H^{*}(G,\mathbb{Z}/2\mathbb{Z}) generated
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A classification of finite primitive IBIS groups with alternating socle J. Group Theory (IF 0.5) Pub Date : 2023-02-22 Melissa Lee, Pablo Spiga
Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , … , ω ℓ ) (\omega_{1},\ldots,\omega_{\ell}) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem
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Character degrees of normally monomial 𝑝-groups of maximal class J. Group Theory (IF 0.5) Pub Date : 2023-02-01 Dongfang Yang, Heng Lv
A finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups
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Conjugacy classes of maximal cyclic subgroups J. Group Theory (IF 0.5) Pub Date : 2023-02-01 Mariagrazia Bianchi, Rachel D. Camina, Mark L. Lewis, Emanuele Pacifici
In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ( G / N ) = η ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial