Recent zbMATH articles in MSC 20Bhttps://www.zbmath.org/atom/cc/20B2021-02-27T13:50:00+00:00WerkzeugSubgroups with no abelian composition factors are not distinguished.https://www.zbmath.org/1453.200032021-02-27T13:50:00+00:00"Chamberlain, Robert"https://www.zbmath.org/authors/?q=ai:chamberlain.robert-fFor a finite group $G$, let $m(G)$ denote the least integer $n$ such that $G$ embeds in the symmetric group of degree $n$. The group $G$ is called exeptional if there exists a normal subgroup of $G$ with $m(G/N)>m(G)$ in which case $N$ is called distinguishable. In the paper under review, the author proves that a subgroup with no abelian composition factors is not distinguishable.
Reviewer: Mohammad-Reza Darafsheh (Tehran)Sharply 2-transitive groups of characteristic 0.https://www.zbmath.org/1453.200042021-02-27T13:50:00+00:00"Rips, Eliyahu"https://www.zbmath.org/authors/?q=ai:rips.eliyahu"Tent, Katrin"https://www.zbmath.org/authors/?q=ai:tent.katrinSummary: We construct sharply 2-transitive groups of characteristic 0 without regular normal subgroups. These groups act sharply 2-transitively by conjugation on their involutions. This answers a long-standing open question.Tamari lattices for parabolic quotients of the symmetric group.https://www.zbmath.org/1453.200512021-02-27T13:50:00+00:00"Mühle, Henri"https://www.zbmath.org/authors/?q=ai:muhle.henri"Williams, Nathan"https://www.zbmath.org/authors/?q=ai:williams.nathan-f|williams.nathan-cSummary: We generalize the Tamari lattice by extending the notions of \(231\)-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group \(\mathfrak{S}_n \). We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.On the cycle index and the weight enumerator.https://www.zbmath.org/1453.111672021-02-27T13:50:00+00:00"Miezaki, Tsuyoshi"https://www.zbmath.org/authors/?q=ai:miezaki.tsuyoshi"Oura, Manabu"https://www.zbmath.org/authors/?q=ai:oura.manabuSummary: In this paper, we introduce the concept of the complete cycle index and discuss a relation with the complete weight enumerator in coding theory. This work was motivated by \textit{P.J. Cameron}'s lecture notes [``Polynomial aspects of codes, matroids and permutation groups'', Preprint, \url{http://www.maths.qmul.ac.uk/ pjc/csgnotes/cmpgpoly.pdf}].Irreducible restrictions of representations of symmetric and alternating groups in small characteristics.https://www.zbmath.org/1453.200172021-02-27T13:50:00+00:00"Kleshchev, Alexander"https://www.zbmath.org/authors/?q=ai:kleshchev.alexander-s"Morotti, Lucia"https://www.zbmath.org/authors/?q=ai:morotti.lucia"Tiep, Pham Huu"https://www.zbmath.org/authors/?q=ai:tiep.pham-huuIn the paper under review, the following problem is investigated: Let \(H\) be the symmetric group or the alternating group on \(n\) letters. The authors classify the pair \((G,V)\), where \(G\) is a subgroup of \(H\) and \(V\) is an FH-module of dimention greater than 1 such that the restriction of \(V\) to \(G\) is irreducible. This has applications to the Aschbacher-Scott programm for maximal subgroupsof the finite classical groups.
Reviewer: Mohammad-Reza Darafsheh (Tehran)The use of permutation representations in structural computations in large finite matrix groups.https://www.zbmath.org/1453.200012021-02-27T13:50:00+00:00"Cannon, John J."https://www.zbmath.org/authors/?q=ai:cannon.john-j"Holt, Derek F."https://www.zbmath.org/authors/?q=ai:holt.derek-f"Unger, William R."https://www.zbmath.org/authors/?q=ai:unger.william-rSummary: We determine the minimal degree permutation representations of all finite groups with trivial soluble radical, and describe applications to structural computations in large finite matrix groups that use the output of the \texttt{CompositionTree} algorithm. We also describe how this output can be used to help find an effective base and strong generating set for such groups. We have implemented the resulting algorithms in \textsf{Magma}, and we report on their performance.Soficity, short cycles, and the Higman group.https://www.zbmath.org/1453.430012021-02-27T13:50:00+00:00"Helfgott, Harald A."https://www.zbmath.org/authors/?q=ai:helfgott.harald-andres"Juschenko, Kate"https://www.zbmath.org/authors/?q=ai:juschenko.kateAt the moment of writing this review, it is not known whether there exists a nonsofic group. The paper is motivated by the question to test this problem asking whether the Higman group (\(H_{4,2}\) in the next definition) is sofic. Here if \(n,m\ge2\), then \(H_{n,m}\) denotes the group generated by elements \(a_1,\dots,a_n\) satisfying the relations \(a_i^{-1}a_{i+1}a_i=a_i^m\) for \(1\le i\le n\) and \(a_n^{-1}a_1a_n=a_1^m\). The authors prove (Theorem~2) that if \(H_{4,m}\) with \(m\ge 2\) is sofic then for every \(\epsilon>0\), there is an \(N>0\) such that for every \(n\ge N\) coprime to \(m\), there is a bijection \(f:\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}\) such that \(f(x+1)=mf(x)\) for at least \((1-\epsilon)n\) elements \(x\) of \(\mathbb{Z}/n\mathbb{Z}\), and \(f(f(f(f(x))))=x\) for all \(x\in\mathbb{Z}/n\mathbb{Z}\). That is, there is a function on \(\mathbb{Z}/n\mathbb{Z}\) that behaves almost like a modular exponential function \(x\mapsto a(x)k^x\) but all its cycles are of length 4 (for connections of such functions with groups \(H_{4,m}\), \(m>2\), consult \textit{L.~Glebsky} [``$p$-quotients of the G.~Higman group'',
Preprint, \url{arXiv:1604.06359}]). The authors were not able to prove that such a function does not exist. Nevertheless they prove (Theorem~1): If \(m,n\ge 2\) are coprime integers and \(f_{m,n} : \mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}: x\mapsto m^x\), then \(f_{n,m}(f_{n,m}(f_{n,m}(x)))=x\) can hold for at most \(o_m(n)\) elements \(x\in \mathbb{Z}/n\mathbb{Z}\), or in other words, an exponential map has cycles of length 3 almost everywhere.
Reviewer: Štefan Porubský (Praha)On the dimension of permutation vector spaces.https://www.zbmath.org/1453.111622021-02-27T13:50:00+00:00"Reis, Lucas"https://www.zbmath.org/authors/?q=ai:reis.lucasSummary: Let \(K\) be a field that admits a cyclic Galois extension of degree \(n\geq 2\). The symmetric group \(S_n\) acts on \(K^n\) by permutation of coordinates. Given a subgroup \(G\) of \(S_n\) and \(u\in K^n\), let \(V_G(u)\) be the \(K\)-vector space spanned by the orbit of \(u\) under the action of \(G\). In this paper we show that, for a special family of groups \(G\) of affine type, the dimension of \(V_G(u)\) can be computed via the greatest common divisor of certain polynomials in \(K[x]\). We present some applications of our results to the cases \(K=\mathbb{Q}\) and \(K\) finite.Corrigendum and addendum to: ``Centralizers of finite subgroups in Hall's universal group''.https://www.zbmath.org/1453.200502021-02-27T13:50:00+00:00"Kegel, Otto H."https://www.zbmath.org/authors/?q=ai:kegel.otto-h"Kuzucuoğlu, Mahmut"https://www.zbmath.org/authors/?q=ai:kuzucuoglu.mahmutSummary: In Hall's universal group every non-trivial conjugacy class satisfies \(CC=U\). Hence generalized version of J. G. Thompson's conjecture is true for every non-trivial conjugacy class \(C\) in \(U\). Moreover Ore's conjecture (every element is a commutator) is true for \(U\) is added to [the authors, ibid. 138, 283--288 (2017; Zbl 1380.20039)]. In [loc. cit., Theorem 2.4] \(C_U(F)/Z(F)\cong U\) is true if \(Z(F)=1\).Erratum to: ``Association schemes for diagonal groups''.https://www.zbmath.org/1453.051382021-02-27T13:50:00+00:00"Cameron, Peter J."https://www.zbmath.org/authors/?q=ai:cameron.peter-j"Eberhard, Sean"https://www.zbmath.org/authors/?q=ai:eberhard.seanSummary: Unfortunately our paper [ibid. 75, Part 3, 357--364 (2019; Zbl 1429.05205)] contains an error in Section 2 which invalidates the main result. The error pertains to the elements \(u_i\) and \(v_i\) defined on page 360. If such elements exist then they are counted as claimed, but they may not exist at
all. We have been unable to fix the error. As a result, the question of AS-freeness of primitive permutation groups of primitive diagonal type remains open.Addendum to: ``Sharply 2-transitive groups of characteristic 0''.https://www.zbmath.org/1453.200052021-02-27T13:50:00+00:00"Scherff, Malte"https://www.zbmath.org/authors/?q=ai:scherff.malte"Tent, Katrin"https://www.zbmath.org/authors/?q=ai:tent.katrinSummary: In this short note we show how to modify the construction of non-split sharply 2-transitive groups of characteristic 0 given in [the authors, ibid. 750, 227--238 (2019; Zbl 1453.20004)] to allow for arbitrary fields of characteristic 0.