Arturo Magidin
A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order $p^{1+\lfloor\frac{c-1}{p-1}\rfloor}$. This is best possible.
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A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.
Let p be a prime number. We give the explicit structure of 2- nilpotent multiplier for each finite 2-generator p-group of class two. Moreover, 2-capable groups in that class are characterized.
This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.
We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.
A presentation as well as a structural description of the automorphism group of a family of 3-generator finite $p$-groups is given, $p$ being an odd prime.
Matrix generators for the general and special linear groups, the symplectic groups and the general and special unitary groups over finite fields. For the most part the generators have been obtained by translating Steinberg's generators for groups of Lie type into matrix form.
The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.
It is shown that there are exactly seventy-eight 3-generator 2- groups of order 2^11 with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.
We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\rm rank}(G_i^{\rm ab})$ in terms of ${\rm rank}(G^{\rm ab})$ in each case.
In this paper we describe the profinite completion of the free solvable group on m generators of solvability length r>1. Then, we show that for m=r=2, the free metabelian group on two generators does not have the Congruence Subgroup Property.
The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.
We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.
We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups
Exponent and number of generators in a finite group
本文给出了有限群生成元个数的一个尖锐上界，该上界由群的阶与指数之比决定。
We deal with two-generator subgroups of PSL(2,C) with real traces of both generators and their commutator. We give discreteness criteria for these groups when at least one of the generators is parabolic. We also present a list of the corresponding orbifolds.
A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that every 2-generator $E$-group is abelian and that all 3-generator $E$-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator $E$-group is abelian. We conjecture that every finite 3-generator $E$-group should be abelian. Moreover we show that the minimum order of a non-abelian $pE$-group is $p^8$ for any odd prime number $p$…
Here we provide three new presentations of Coxeter groups type $A$, $B$, and $D$ using prefix reversals (pancake flips) as generators. We prove these presentations are of their respective groups by using Tietze transformations on the presentations to recover the well known presentations with generators that are adjacent transpositions.
We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C_1,...,C_n and the minimal number of generators of G is n.
We show that non-abelian two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded free groups. This provides an alternate proof of a theorem of A. Baudisch: that all two-generator subgroups are free or free abelian. Additionally, it shows that they are quasi-isometrically embedded. Our theorem also gives a method for detecting groups that are not isomorphic to a subgroup of any RAAG. We present some counterexamples in subgroups with more than two generators.
We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this we gather together known discreteness criteria for two-generator groups and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a $π$-loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such groups.
In this paper, we obtain Groebner-Shirshov (non-commutative Gröbner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word" $δ$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form in the braid groups, and thus a new algorithm for solving the word problem in these groups.
We prove that if $G$ is a transitive permutation group, then $d(G)\log{|G|}/n^{2}$ tends to $0$ as $n$ tends to $\infty$.
In this paper, we extend the method in [FFLP] to obtain the generators of the Picard modular groups $\mathbf{PU}(2,1;\mathcal {O}_d)$ with $d=3,7,11$.
Let $G$ be the free two step nilpotent Lie group on three generators and let $L$ be a sub-Laplacian on it. We compute the spectral resolution of $L$ and prove that the operators arising from this decomposition enjoy a Tomas-Stein type estimate.
In this paper we give necessary and sufficient conditions for discreteness of a group generated by a hyperbolic element and an elliptic one of odd order. This completes the classification of discrete groups with non-$π$-loxodromic generators in the class of two-generator groups with real parameters. The criterion is given also as a list of all parameters that correspond to discrete groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold has real parameters.
We compute the numbers g(n,2,2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}.
For a positively presented group G, we provide a criterion for two tuples of positive group generators of G to be Hurwitz equivalent or Hurwitz-conjugation equivalent. We also present an algorithmic approach to solve the Hurwitz equivalence and the Hurwitz search problems by using the word reversing method.
We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $γ_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.
In this paper, we give a Gröbner-Shirshov basis of the braid group $B_{n+1}$ in Adyan-Thurston generators. We also deal with the braid group of type $\bf{B}_{n}$. As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup $B^+_{n+1}$ is the subsemigroup in $B_{n+1}$.
We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.
We generalize the Plesken-Fabiańska $\mathrm{L}_2$-quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the invariant ring of $\mathrm{GL}(2, K)$ on $m$ copies of $\mathrm{SL}(2, K)$ by simultaneous conjugation. By giving this description, we generalize and simplify some of the known results in invariant theory. An implementation of the algorithm is available in the computer algebra system Magma.
A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which may be checked explicitly for any specific instance of the problem. We use this restatement to derive some known results, and prove new ones. Among them, we reduce the general problem to an oft-considered special case, and prove that a 3-generated group of class 2 and exponent $p$ is either cyclic or capable.
For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.
It was conjectured by Tits that the only relations amongst the squares of the standard generators of an Artin group are the obvious ones, namely that a^2 and b^2 commute if ab=ba appears as one of the Artin relations. In this paper we prove Tits' conjecture for all Artin groups. More generally, we show that, given a number m(s)>1 for each Artin generator s, the only relations amongst the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba appears amongst the Artin relations.
A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order $p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, {\em On the non abelian tensor square and capability of groups of order $p^2q$}, Arch. Math., \textbf{97} (2011), 299--306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order $p^3$, which extends Proposition 2.10 of S. M. Jafarian Amiri, H. Ma…
It is well known that, in general, the set of commutators of a group $G$ may not be a subgroup. Guralnick showed that if $G$ is a finite $p$-group with $p\ge 5$ such that $G'$ is abelian and $3$-generator, then all the elements of the derived subgroup are commutators. In this paper, we extend Guralnick's result by showing that the condition of $G'$ to be abelian is not needed. In this way, we complete the study of this property in finite $p$-groups in terms of the number of generators of the derived subgroup. We will also see that the same result is true when the action of $G$ on $G'$ is unise…
Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups.
In this paper we give some conditions in which a direct product of groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. Moreover, we give some conditions in which a direct product of a finite family of pairs of groups is capable if and only if each of its factors is a capable pair of groups.
A continuum of non-isomorphic 3-generator groups with probabilistic law $x^n=1$
本研究构造了一族连续统的非同构3生成元群，利用$n$阶循环群的$n$-周期积与2生成元相对自由群，证明群中$x^n=1$以概率1成立但非普遍成立，解决了群恒等式的概率满足与普遍满足关系的问题。
We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.
Lifting Generators in Connected Lie Groups
本文证明连通李群的生成元提升问题可归结为阿贝尔化情形，并据此得出完美李群满足 Gaschütz 引理，同时给出非完美情形的 Gaschütz 秩界及精确计算。
A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.
Topology of the space of bi-orderings of a free group on two generators
本文研究二元生成自由群双序空间的拓扑性质，证明了其双序空间$O(F_2)$与左序空间一样不存在孤立点。
Finding cycle types in permutation groups with few generators
本文研究了置换群中寻找特定圈型置换的问题，证明了该问题对循环置换群可在对数空间求解，而对2生成的阿贝尔置换群及无不动点判定均为NP完全。
We construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \dots$, where each $G_m$ has a specific presentation $G_m=\langle a, b \, | \, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \cdots =1 \rangle$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\langle m \rangle,m+1,m]$ in continued fraction expansion for every integer $i \ge 1$.
In this paper, we characterize the finite groups $G$ of even order with the property that for any involution $x$ and element $y$ of $G$, $\langle x, y \rangle$ is isomorphic to one of the following groups: $\mathbb{Z}_2,$ $\mathbb{Z}_2^2$, $\mathbb{Z}_4$, $\mathbb{Z}_6$, $\mathbb{Z}_2 \times \mathbb{Z}_4$, $\mathbb{Z}_2 \times\mathbb{Z}_6$ and $A_4$. As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree $3$ have integral spectrum.
Let $G$ be a finite $p$-group whose derived subgroup $G'$ can be generated by $2$ elements. If $G'$ is abelian, Guralnick proved that every element of $G'$ is a commutator. In this paper, we prove that the condition that $G'$ should be abelian is not needed. Even more, we prove that every element of $G'$ is a commutator of the form $[x,g]$ for a fixed $x\in G$.
Let $G \simeq M \rtimes C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful module over a suitable quotient $R$ of the integral group ring of $C$. When $C$ is infinite, we show that the Nielsen equivalence classes of the generating $n$-tuples of $G$ correspond bijectively to the orbits of unimodular rows in $M^{n -1}$ under the action of a subgroup of $GL_{n - 1}(R)$. Making no assumption on the cardinality of $C$, we exhibit a comp…
A group in which every element commutes with its endomorphic images is called an $E$-group. Our main result is that all 3-generator $E$-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian $E$-group is four.
We calculate explicitly the normal zeta function of the free class two nilpotent group on four generators. This has Hirsch length ten.
The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring-Martin-Tan group if the equality holds for the group $G$. We give a method for constructing Gehring-Martin-Tan groups with a generator of order four and present some examples. These groups arise as groups of finite volume hyperbolic 3-orbifolds.
We describe generators and defining relations for the commutator subgroup of topological full groups of minimal subshifts. We show that the word problem in a topological full group is solvable if and only if the language of the underlying subshift is recursive.
We show that the holomorph of the free group on two generators satisfies the Farrell-Jones Fibered Isomorphism Conjecture. As a consequence, we show that the lower K-theory of the above group vanishes.
According to the Tits conjecture proved by Crisp and Paris, [CP], the subgroups of the braid group generated by proper powers of the Artin elements are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroups generated by powers of the band generators is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with expo…
The sister of Eisenstein-Picard modular group was described in [8]. In this paper we give a similar definition of the sister of the Euclidean-Picard modular group and find its generators by using a geometric method.
We completely elucidate the relationship between two invariants associated with an ergodic probability measure-preserving (pmp) equivalence relation, namely its cost and the minimal number of topological generators of its full group. It follows that for any free pmp ergodic action of the free group on $n$ generators, the minimal number of topological generators for the full group of the action is $n+1$, answering a question of Kechris.
Little is known about the generators system of the higher dimensional Picard modular groups. In this paper, we prove that the higher dimensional Eisenstein--Picard modular group $\mathbf{PU}(3,1;\mathbb{Z}[ω_3])$ in three complex dimensions can be generated by four given transformations.
Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is linear. This result gives a large linear subgroup of Aut(F_3). A second application is that the mapping class group for genus one surfaces with two punctures is linear.
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