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. Algebra163 (1994), 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set 𝐵 in the positive

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

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

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

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.

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

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

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).

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.

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

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

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.

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 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

New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups Xni⁢(q) (i∈I) for X∈{PGL,PSp,PGO(ε),PGU} (ε=±) along an ultrafilter 𝒰 on the index set I for which ni→𝒰∞ determines the type X and the field size q up to the possible isomorphism of a metric ultraproduct of groups PSpni⁡(q) and a metric ultraproduct

We explore conditions for the Frobenius–Wielandt morphism to commute with the operations of deflation and tensor induction on the Burnside ring. In doing this, we review the commutativity with induction. The techniques used for induction and tensor induction no longer work for deflation, so, in this case, we make use of tools coming from the theory of biset functors.

Let X be a finite type An2-polyhedron, n≥2. In this paper, we study the quotient group ℰ⁢(X)/ℰ*⁢(X), where ℰ⁢(X) is the group of self-homotopy equivalences of X and ℰ*⁢(X) the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as ℰ⁢(X) or ℰ⁢(X)/ℰ*⁢(X) for X an An2-polyhedron, n≥3, and specific results are obtained for

Let w be a group-word. For a group G, let Gw denote the set of all w-values in G and w⁢(G) the verbal subgroup of G corresponding to w. The word w is semiconcise if the subgroup [w⁢(G),G] is finite whenever Gw is finite. The group G is an FC⁢(w)-group if the set of conjugates xGw is finite for all x∈G. We prove that if w is a semiconcise word and G is an FC⁢(w)-group, then the subgroup [w⁢(G),G] is

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 K1-functor under the stable rank condition.

Given a group action on a finite-dimensional CAT⁢(0) cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser satisfies the strong Tits alternative. We use it to prove that all Artin groups of type FC satisfy the strong Tits alternative.

Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that (k⁢N)⁢c is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a (G,N,c)-Brauer pair (R,fR) consists

We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut⁢(G) on G is bounded from above by a constant. Our main results are the following: Firstly, a finite group G only admits Aut⁢(G)-orbits of length at most 3 if and only if G is cyclic of one of the orders 1, 2, 3, 4 or 6, or G is the Klein four group or the symmetric group of degree 3

Let G be a group, x,y∈G, and H is a normal closure in G of the subgroup generated by all commutators of x,y that include more than two occurrences of y. Our main result is a parametrization of the uncollected part of Hall’s collection formula for (x⁢y)n and two different collection formulas modulo H in an explicit form. In particular, we found explicit expressions for the exponents of the commutators

In this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank

In this note, we present an improvement on the large orbit result of Halasi and Podoski, and then answer an open question raised in [X. Chen, J. P. Cossey, M. Lewis and H. P. Tong-Viet, Blocks of small defect in alternating groups and squares of Brauer character degrees, J. Group Theory 20 2017, 6, 1155–1173].

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

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

For a saturated fusion system ℱ on a p-group S, we study the Burnside ring of the fusion system B⁢(ℱ), as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring B⁢(S). We give criteria for an element of B⁢(S) to be in B⁢(ℱ) determined by the ℱ-automorphism groups of essential subgroups of S. When ℱ is the fusion system induced by a finite group G with S as a Sylow p-group

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

Let G be a finite group, let b be a G-invariant block with defect group Q of the normal subgroup H of G, and let b′∈Z⁢(𝒪⁢NH⁢(Q)) be the Brauer correspondent of b. We show that the bijection between the blocks of G covering b and the blocks of NG⁢(Q) covering b′, induced by a G/H-graded basic Morita equivalence between the block extensions b⁢𝒪⁢G and b′⁢𝒪⁢NG⁢(Q), coincides with the Harris–Knörr correspondence

Journal Name: Journal of Group Theory Volume: -1 Issue: ahead-of-print

Journal Name: Journal of Group Theory Volume: -1 Issue: ahead-of-print

The Jordan–Hölder theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan–Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan–Hölder type theorem for supercharacter theories of finite groups, which generalizes

Let n be a positive integer. We say that a group G is an (n+12)-Engel group if it satisfies the law [x,yn,x]=1. The variety of (n+12)-Engel groups lies between the varieties of n-Engel groups and (n+1)-Engel groups. In this paper, we study these groups, and in particular, we prove that all (4+12)-Engel {2,3}-groups are locally nilpotent. We also show that if G is a (4+12)-Engel p-group, where p≥5 is

Let R be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over R. Our main result states that if G is a hereditary group over R, then a unital algebra isomorphism between group algebras R⁢G≅R⁢H implies a group isomorphism G≅H for every finite group H. As application, we study the modular isomorphism problem, which is the isomorphism problem

A group G has the R∞-property if the number R⁢(φ) of twisted conjugacy classes is infinite for any automorphism φ of G. For such a group G, the R∞-nilpotency degree is the least integer c such that G/γc+1⁢(G) still has the R∞-property. In this paper, we determine the R∞-nilpotency degree of all Baumslag–Solitar groups.

If 𝔛 is a class of groups, define a sequence of group classes 𝔛1,𝔛2,…,𝔛k,… by putting 𝔛1=𝔛 and choosing 𝔛k+1 as the class of all groups whose non-permutable subgroups belong to 𝔛k. In particular, if 𝔄 is the class of abelian groups, 𝔄2 is the class of quasimetahamiltonian groups, i.e. groups whose non-permutable subgroups are abelian. The aim of this paper is to study the structure of 𝔛k-groups

We prove that each exotic fusion system ℱ on a Sylow p-subgroup of G2⁢(p) for an odd prime p with 𝒪p⁢(ℱ)=1 is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.

Let κ be a characteristic p finite field of q elements and 𝔑κ the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of 𝔑κ a type. We establish an upper bound B⁢(q;l,m) on the number of conjugacy classes of order p2 torsion elements u of 𝔑κ of type 〈l,m〉. In the case where l