If a finite group G has a presentation with d generators and r relations, it is well-known that r - d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations
We implement GAP functions about groups with action on itself and investigate some basic properties of small groups with action on itself of order $<32$.
We observe that the results of Feighn-Handel and Geoghegan-Mihalik-Sapir-Wise about coherence and Hopficity of mapping tori of injective endomorphisms of free groups extend to non-injective endomorphisms.
We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The first-one concerns the non-superstability of such groups in most cases, and the second-one concerns the connectedness of most free products of groups.
Matthew Brin and Patrick Dehornoy independently discovered a braided version BV of Thompson's group V. In this paper, we discuss some properties of BV that might make the group interesting for group based cryptography. In particular, we show that BV does not admit a non-trivial linear representation.
We describe $\{2,3\}$-groups in which the order of a product of any two elements of orders at most $4$ does not exceed $9$ and the centralizer of every involution is a locally cyclic $2$-subgroup. In particular, we will prove that these groups are locally finite.
It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories.
The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees $2n$ and $2n-1$ to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for $p+1$ and $p-1$ whenever $p \geq 11$ is a prime number.
Examples are given of profinite groups that are not strongly complete, and have other `bad' properties, yet have only finitely many open subgroups of each finite index. It is shown that a profinite group with the latter property must be finite if it has finite exponent. The problem of characterizing strongly complete groups in terms of their power subgroups is discussed.
Let ${\cal N}_c$ be the variety of nilpotent groups of class at most $c\ \ (c\geq 2)$ and $G=Z_r\oplus Z_s $ be the direct sum of two finite cyclic groups. It is shown that if the greatest common divisor of $r$ and $s$ is not one, then $G$ does not have any ${\cal N}_c$-covering group for every $c\geq 2$. This result gives an idea that Lemma 2 of J.Wiegold [6] and Haebich's Theorem [1], a vast generalization of the Wiegold's Theorem, can {\it not} be generalized to the variety of nilpotent groups of class at most $c\geq 2$.
A subset $S$ of a group $G$ invariably generates $G$ if $G$ is generated by $\{ s^g(s) | s\in S\} $ for any choice of $g(s)\in G, s\in S$. In case $G$ is topological one defines similarly the notion of topological invariable generation. A topological group $G$ is said to be $\mathcal{TIG}$ if it is topologically invariably generated by some subset $S\subseteq G$. In this paper we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group $\mathrm{SL}_{2}(\mathbb{R})$ and the automorphism group of a r…
Let $F$ be a free group of positive, finite rank and let $Φ\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_Φ\mathbb Z$ is strongly thick of order $η$, where $η$ is the rate of polynomial growth of $ϕ$. This fact is implicit in work of Macura, but her work predates the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth and thickness explicit. Our result combines with a result independently due to Dahmani-Li, Gautero-Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgr…
Let $G$ be an arbitrary group. We show that if the Fitting subgroup of $G$ is nilpotent then it is definable. We show also that the class of groups whose Fitting subgroup is nilpotent of class at most $n$ is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given.
For a product of Hadamard spaces $X=X_1\times X_2$ on which some group $G\subset Is(X_1)\times Is(X_2)$ acting, G. Link arXiv:1107.3755v1 introduced the growth rate $δ_θ$ of slope $θ$ to construct a $G-$invariant $(b,θ)$-density. First, we show that $δ_θ$ is continuous in the slope $θ$ as the above group action with some mild condition. Second, we give a negative answer to a question raised by G. Link in \cite{1} in general. And further results about the question are discussed. In the end, we can extent the $(b,θ)$-densities in some reasonable condition.
School of Mathematics and Statistics, Henan University of Technology（河南工业大学数学与统计学院）；Faculty of Mathematics, K. N. Toosi University of Technology（K. N. 托伊西理工大学数学学院）
We provide an infinite family of left-ordered groups, all of which have a positive cone that is finitely generated as a semigroup. This family includes the Klein bottle group and the braid group B_3.
We prove that commutator subgroups of topological full groups arising from minimal subshifts have exponential growth. We also prove that the measurable full group associated to the countable, measure-preserving, ergodic and hyperfinite equivalence relation is topologically generated by two elements.
We improve a result of Prokhorov and Shramov on the rank of finite $p$-subgroups of the birational automorphism group of a rationally connected variety. Known examples show that they are sharp in many cases.
Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $Γ$ with respect to $H$ admit a natural semigroup structure. For any compact group $K$ the semigroup $Γ$ acts in the space $L^2$ on the product of $m$ copies of $K$
We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, real algebraic groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper, we determine the possible values of $m\in \mathbb{N}$ for which the above problem has a solution. We also show that there is an upper bound on the maximal order of an element in $G$. As an illustration, we apply our results to the group $\Sp(4,\mathbb{Z})$ and determine the possible orders of elements in it. Finally, we use a presentation of $\Sp(4,\mathbb{Z}…
The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length fun…
In this short note we survey some results about the fundamental group of a compact negatively curved manifold. In particular, we review a theorem of Gusevskij, it states that the fundamental group of a compact negatively curved manifold does not belong to $\mathcal{C},$ where $\mathcal{C}$ is the smallest class of groups that contains all amenable groups and is closed under free products and finite extensions.
We construct three groups $Λ_1$, $Λ_2$, $Λ_3$, which can all be decomposed as amalgamated products $F_9 \ast_{F_{81}} F_{9}$ and have very few normal subgroups of finite or infinite index. Concretely, $Λ_1$ is a simple group, $Λ_2$ is not simple but has no non-trivial normal subgroup of infinite index, and $Λ_3$ is not simple but has no proper subgroup of finite index.
We notice that the class of nontrivial groups without proper subgroups of finite index is not elementary, because some groups in this class, such as $\mathbb Q*\mathbb Q$, have ultrapowers that map homomorphically onto $\mathbb Z/p\mathbb Z$ for every prime $p$. Also, some ultrapowers of certain simple groups map homomorphically onto $\mathbb Z/2\mathbb Z$.
The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_\infty$-property and are not co-Hopfian for $n \ge 3$.
We observe that Clay-Rolfsen's obstruction of bi-orderability, which uses the classical Alexander polynomial, is not strengthened by using the twisted Alexander polynomials for finite representations unlike many known applications of the Alexander polynomial. This is shown by studying the maximal ordered abelian quotient of bi-ordered groups.
In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.
A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.
We discuss the Bohr compactification of a pseudofinite group, motivated by a question of Boris Zilber. Basically referring to results in the literature we point out (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the "definable" Bohr compactification of any pseudofinite group G, relative to a nonstandard model of set theory in which it is definable, is commutative-by-profinite.
The present paper is a note on the tensor degree of finite groups, introduced recently in literature. This numerical invariant generalizes the commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the tensor and the commutativity degree of finite groups, and, indirectly, structural properties will be discussed.
Let X, Y, and Z be topological modules over a topological ring $R$. In the first part of the paper, we introduce three different classes of bounded bigroup homomorphisms from $X\times Y$ into $Z$ with respect to the three different uniform convergence topologies. We show that these spaces form again topological modules over $R$. In the second part, we characterize bounded sets in the arbitrary product of topological groups with respect to the both product and box topologies.
We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy cl…
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.
Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed.
It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to any set of generators. Partial results in this direction were obtained by Šunić and by Warshall. We improve these results by showing that abelian groups only have finitely many dead ends and that groups with more than one end (in the sense of Hopf and Freudenthal) have only dead ends of bounded depth. Only few examples of groups with unbounded dead end depth …
By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank $\geq 2$ equals 1. This answers a question of J-F. Lafont and S. Prassidis.
Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs
Scuola Galileiana di Studi Superiori, Università di Padova（高等研究伽利略学院，帕多瓦大学）；Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova（帕多瓦大学，“图利奥·莱维-奇维塔”数学系）；University of Milano-Bicocca（米兰比可卡大学）
The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.
We prove that for a weakly mixing algebraic action $σ: G\curvearrowright(X,ν)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $σ^2, σ^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when $G$…
We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbolΛ(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the con…
