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Carlos A. M. André, João Dias
May 26, 2021
Article number: 000010151520190084

### Abstract

We consider smooth representations of the unit group G=A×G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

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Wenhao Wang
May 19, 2021
Article number: 000010151520200182

### Abstract

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

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Tamar Bar-On
May 18, 2021
Article number: 000010151520210001

### Abstract

We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

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Genildo de Jesus Nery
May 13, 2021
Article number: 000010151520200175

### Abstract

In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n⩽21n\leqslant 21, then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

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Arunava Mandal
May 13, 2021
Article number: 000010151520200152

### Abstract

Let 𝐺 be a complex algebraic group defined over ℝ, which is not necessarily Zariski-connected. In this article, we study the density of the images of the power maps g→gkg\to g^{k}, k∈Nk\in\mathbb{N}, on real points of 𝐺, i.e., G(R)G(\mathbb{R}) equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of G(R)G(\mathbb{R}). We also characterize surjectivity of the power map for a disconnected group G(R)G(\mathbb{R}). The results are applied in particular to describe the image of the exponential map of G(R)G(\mathbb{R}).

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Hamza Alzaareer
May 12, 2021
Article number: 000010151520180200

### Abstract

We study the existence of Lie group structures on groups of the form Ck(M,K)C^{k}(M,K), where 𝑀 is a non-compact smooth manifold with rough boundary and 𝐾 is a, possibly infinite-dimensional, Lie group. Motivated by introducing this new class of infinite-dimensional Lie groups, we obtain a new version of the fundamental theorem for Lie algebra-valued functions.

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Yan-Quan Feng, István Kovács
April 7, 2021
Article number: 000010151520200094

### Abstract

In this paper, we characterize the finite groups 𝐺 of even order with the property that, for any involution 𝑥 and element 𝑦 of 𝐺, ⟨x,y⟩\langle x,y\rangle is isomorphic to one of the following groups: Z2\mathbb{Z}_{2}, Z22\mathbb{Z}_{2}^{2}, Z4\mathbb{Z}_{4}, Z6\mathbb{Z}_{6}, Z2×Z4\mathbb{Z}_{2}\times\mathbb{Z}_{4}, Z2×Z6\mathbb{Z}_{2}\times\mathbb{Z}_{6} and A4A_{4}. As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree 3 have integral spectrum.

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Guohua Qian, Yuanyang Zhou
March 31, 2021
Article number: 000010151520200090

### Abstract

Let 𝐺 be a finite group and 𝑝 a prime. We define the codegree of χ∈Irr(G)\chi\in\mathrm{Irr}(G) by cod(χ)=|G:kerχ|/χ(1)\operatorname{cod}(\chi)=\lvert G:\ker\chi\rvert/\chi(1) and define cp(G)=max{logp(cod(χ))p∣χ∈Irr(G)}c_{p}(G)=\max\{\log_{p}(\operatorname{cod}(\chi))_{p}\mid\chi\in\mathrm{Irr}(G)\}. In this paper, we show that |G/Op′p(G)|p≤pcp(G)\lvert G/O_{p^{\prime}p}(G)\rvert_{p}\leq p^{c_{p}(G)} for all finite groups 𝐺 and characterize the finite groups 𝐺 with cp(G)≤1c_{p}(G)\leq 1.

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Nariel Monteiro
March 16, 2021
Article number: 000010151520200148

### Abstract

Let O2\mathcal{O}_{2} and O2′\mathcal{O}^{\prime}_{2} be two distinct finite local rings of length two with residue field of characteristic 𝑝. Let G(O2)\mathbb{G}(\mathcal{O}_{2}) and G(O2′)\mathbb{G}(\mathcal{O}^{\prime}_{2}) be the groups of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G×Fq\mathbb{G}\times\mathbb{F}_{q} or G=GLn\mathbb{G}=\operatorname{GL}_{n}. We prove that there exists an isomorphism of group algebras KG(O2)≅KG(O2′)K\mathbb{G}(\mathcal{O}_{2})\cong K\mathbb{G}(\mathcal{O}^{\prime}_{2}), where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

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Egle Bettio
March 12, 2021
Article number: 000010151520200129

### Abstract

In this paper, we prove that if 𝐺 is a group generated by elements of order two with the property that the product of any two such elements has order 1, 2, 3 or 5 with all possibilities occurring, then G≃A5G\simeq A_{5} or G≃PSU(3,4)G\simeq\mathrm{PSU}(3,4). This provides an affirmative answer to Problem 19.36 in the Kourovka notebook.

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Francesco Matucci, Pedro V. Silva
March 11, 2021
Article number: 000010151520200034

### Abstract

In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group Lk=Zk≀Z\mathcal{L}_{k}=\mathbb{Z}_{k}\wr\mathbb{Z}.

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Maria Alicia Aviño, Phill Schultz, Marcos Zyman
March 6, 2021
Article number: 000010151520200106

### Abstract

Let 𝐺 be a bounded abelian 𝑝-group, with automorphism group Aut(G)\operatorname{Aut}(G). Whenever 𝐺 satisfies certain conditions, we determine the upper central series and nilpotency class of the maximal normal 𝑝-subgroup of Aut(G)\operatorname{Aut}(G).

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Denis Osin
February 16, 2021
Article number: 000010151520200150

### Abstract

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 UV\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 obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in UV\mathcal{UV}.

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James Williams
February 2, 2021
Article number: 000010151520200197

### Abstract

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 on the order of commutators in powerful 𝑝-groups to this larger family of groups. We also obtain a bound on the number of generators of a subgroup of a quasi-powerful 𝑝-group, expressed in terms of the number of generators of the group, and we give an example which demonstrates this bound is close to best possible.

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Daniel El-Baz, Carlo Pagano
February 2, 2021
Article number: 000010151520200066

### Abstract

We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

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Isabel Fernández Martínez, Denis Serbin
February 2, 2021
Article number: 000010151520200133

### Abstract

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 hyperbolic groups and finitely generated subgroups of limit groups.

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Manuel Bodirsky, Bertalan Bodor
January 20, 2021
Article number: 000010151520180220

### Abstract

Let Kexp+\mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most cndncn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c,dc,d with d<1d<1. We show that Kexp+\mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that Kexp+\mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from Kexp+\mathcal{K}_{{\operatorname{exp}}{+}}. We also show that Thomas’ conjecture holds for Kexp+\mathcal{K}_{{\operatorname{exp}}{+}}: all structures in Kexp+\mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

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John J. Ballantyne, Peter J. Rowley
January 14, 2021
Article number: 000010151520200120

### Abstract

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=2aq=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 for the number of elements in 𝑋 which have odd order product with 𝑡, in terms of 𝑞.

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James B. Wilson
January 12, 2021
Article number: 000010151520200121

### Abstract

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 point pertains to at least a positive logarithmic proportion of all finite groups.

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Stefanos Aivazidis, Inna N. Safonova, Alexander N. Skiba
December 23, 2020
Article number: 000010151520200149

### Abstract

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{\mathfrak{F}}-subnormal in the sense of Kegel , or K -𝔉{\mathfrak{F}}-subnormal in G , if there is a subgroup chain A=A0⩽A1⩽⋯⩽An=G{A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G} such that either Ai-1⊴Ai{A_{i-1}\trianglelefteq A_{i}} or Ai/(Ai-1)Ai∈𝔉{A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F}} for all i=1,…,n{i=1,\ldots,n}. In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let F{\mathfrak{F}} be a hereditary saturated formation containing all nilpotent groups, and let S be a K -F{\mathfrak{F}}-subnormal subgroup of G . If ZF(E)=1{\mathbf{Z}_{\mathfrak{F}}(E)=1} for every subgroup E of G such that S⩽E{S\leqslant E}, then CG(D)⩽D{\mathbf{C}_{G}(D)\leqslant D}, where D=SF{D=S^{\mathfrak{F}}} is the F{\mathfrak{F}}-residual of S .

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Ulderico Dardano, Fausto De Mari
October 22, 2020
Article number: 000010151520200076

### Abstract

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.

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Enrique Miguel Barquinero, Lorenzo Ruffoni, Kaidi Ye
January 5, 2020
Article number: 000010151520200124

### Abstract

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.