Some considerations on Hydra groups and a new bound for the length of words

Abstract After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins.

Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

In the present paper we show that it is possible to obtain the well known Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^4=(ZX)^4=(XY)^4=1 \rangle $ of order $16$ as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere $S^3$. The first of these spaces of orbits is realized via an action of the quaternion group $Q_8$ on $S^3$; the second one via an action of the cyclic group of order four $\mathbb{Z}(4)$ on $S^3$. We deduce a result of decomposition of $P$ of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.

A note on relative isoclinism classes of compact groups

A note on the exterior centralizer

The notion of the exterior centralizer \({C_G^{^\wedge}(x)}\) of an element x of a group G is introduced in the present paper in order to improve some known results on the non-abelian tensor product of two groups. We study the structure of G by looking at that of \({C_G^{^\wedge}(x)}\) and we find some bounds for the Schur multiplier M(G) of G.

Commuting powers and exterior degree of finite groups

In [P. Niroomand, R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335-343] it is introduced a group invariant, related to the number of elements $x$ and $y$ of a finite group $G$, such that $x \wedge y = 1_{G \wedge G}$ in the exterior square $G \wedge G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m \wedge k$ of $H \wedge K$ such that $h^m \wedge k = 1_{H \wedge K}$, where $m \ge 1$ and $H$ and $K$ are arbitrary subgroups of $G$.

On the WGSC Property in Some Classes of Groups

The property of quasi-simple filtration (or qsf) for groups has been introduced in literature more than 10 years ago by S. Brick. This is equivalent, for groups, to the weak geometric simple connectivity (or wgsc). The main interest of these notions is that there is still not known whether all finitely presented groups are wgsc (qsf) or not. The present note deals with the wgsc property for solvable groups and generalized FC-groups. Moreover, a relation between the almost-convexity condition and the Tucker property, which is related to the wgsc property, has been considered for 3-manifold groups.

The probability that $x$ and $y$ commute in a compact group

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…

A mathematical description of glitches in neutron stars

In a pulsar, there are gaps and difficulties in our knowledge of glitches, mainly because of the absence of information about the physics of the matter of the star. This has motivated several authors to suggest dynamical models that interpret most of the astronomical data. Many predictions are based on the assumption that the inner part is analogous to the structure of matter of superfluids. Here, we illustrate a new mathematical model, partially inspired by the dynamics of superfluid helium. We obtain two evolution equations for the angular velocities (of the crust and of superfluid), which are supported by another evolution equation for the average vortex line length per unit volume. This…

A probabilistic meaning of certain quasinormal subgroups

The role of the cyclic quasinormal subgroups has been recently described in groups both finite and infinite by S.Stonehewer and G.Zacher. This role can be better analyzed in the class of compact groups, obtaining restrictions for the probability that two randomly chosen elements commute. Mathematcs Subject Classification: 20D60, 20P05, 20D08

A Survey on Just-Non-X Groups

Let be a class of groups. A group which does not belong to but all of whose proper quotient groups belong to is called just-non- group. The present note is a survey of recent results on the topic with a special attention to topological groups.

Atmospheric Background Measurement in the 300–400 nm Band with a Balloon Borne Experiment during a Nocturnal Flight

The balloon borne experiment, named BABY (BAckground BYpass) belongs to a wider program, AIRWATCH-OWL, intended for the observation of high energy Cosmic Rays from space, detecting the faint UV fluorescence light emitted by the atmospheric Nitrogen as final result of a complex hadronic cascade. In this framework, one of the fundamental information concern the knowledge of the background level. This is one of the main parameters that contribute to the sensitivity of any kind of instrument. The apparatus used for the BABY experiment was designed and completely built at the IFCAI-CNR in Palermo. The instrument is composed by two filtered and collimated photomultipliers (PMT) that detect the UV…

$n$-th relative nilpotency degree and relative $n$-isoclinism classes

P. Hall introduced the notion of isoclinism between two groups more than 60 years ago. Successively, many authors have extended such a notion in different contexts. The present paper deals with the notion of relative n-isoclinism, given by N. S. Hekster in 1986, and with the notion of n-th relative nilpotency degree, recently introduced in literature.

THE PROBABILITY THAT AND COMMUTE IN A COMPACT GROUP

AbstractIn a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian,…

A note on the Schur multiplier of a nilpotent Lie algebra

For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.

Groups described by element numbers

Abstract Let G be a finite group and L e ( G ) = { x ∈ G ∣ x e = 1 } $L_e(G)=\lbrace x \in G \mid x^e=1\rbrace $ , where e is a positive integer dividing | G | $\vert G\vert $ . How do bounds on | L e ( G ) | $\vert L_e(G)\vert $ influence the structure of G? Meng and Shi [Arch. Math. (Basel) 96 (2011), 109–114] have answered this question for | L e ( G ) | ≤ 2 e $\vert L_e(G)\vert \le 2e$ . We generalize their contributions, considering the inequality | L e ( G ) | ≤ e 2 $\vert L_e(G)\vert \le e^2$ and finding a new class of groups of whose we study the structural properties.

On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335--343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.

A restriction on the schur multiplier of nilpotent lie algebras

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.

On the exterior degree of the wreath product of finite abelian groups

The exterior degree $d^\wedge(G)$ of a finite group $G$ has been recently introduced by Rezaei and Niroomand in order to study the probability that two given elements $x$ and $y$ of $G$ commute in the nonabelian exterior square $G \wedge G$. This notion is related with the probability $d(G)$ that two elements of $G$ commute in the usual sense. Motivated by a paper of Erovenko and Sury of 2008, we compute the exterior degree of a group which is the wreath product of two finite abelian $p$--groups ($p$ prime). We find some numerical inequalities and study mostly abelian $p$-groups.

On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank

We have recently shown that pseudo-bosonic operators realize concrete examples of finite dimensional nilpotent Lie algebras over the complex field. It has been the first time that such operators were analyzed in terms of nilpotent Lie algebras (under prescribed conditions of physical character). On the other hand, the general classification of a finite dimensional nilpotent Lie algebra $\mathfrak{l}$ may be given via the size of its Schur multiplier involving the so-called corank $t(\mathfrak{l})$ of $\mathfrak{l}$. We represent $\mathfrak{l}$ by pseudo-bosonic ladder operators for $t(\mathfrak{l}) \le 6$ and this allows us to represent $\mathfrak{l}$ when its dimension is $\le 5$.

Some algebraic and topological properties of the nonabelian tensor product

Several authors investigated the properties which are invariant under the passage from a group to its nonabelian tensor square. In the present note we study this problem from the viewpoint of the classes of groups and the methods allow us to prove a result of invariance for some geometric properties of discrete groups.

A description of pseudo-bosons in terms of nilpotent Lie algebras

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

Some criteria for detecting capable Lie algebras

Abstract In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011) 1293–1297], we classify all capable nilpotent Lie algebras of finite dimension possessing a derived subalgebra of dimension one. Indirectly, we find also a criterion for detecting noncapable Lie algebras. The final part contains a construction, which shows that there exist capable Lie algebras of arbitrary big corank (in the sense of Berkovich–Zhou).

On minimal non-PC-groups

On dit qu'un groupe G est un PC-groupe, si pour tout x ∈ G, G/C G (x G ) est une extension d'un groupe polycyclique par un groupe fini. Un non-PC-groupe minimal est un groupe qui n'est pas un PC-groupe mais dont tous les sous-groupes propres sont des PC-groupes. Notre principal resultat est qu'un non-PC-groupe minimal ayant un groupe quotient fini non-trivial est une extension cyclique finie d'un groupe abelien divisible de rang fini.

On $MC$-hypercentral triply factorized groups

A group G is called triply factorized in the product of two subgroups A, B and a normal subgroup K of G ,i fG = AB = AK = BK. This decomposition of G has been studied by several authors, investigating on those properties which can be carried from A, B and K to G .I t is known that if A, B and K are FC-groups and K has restrictions on the rank, then G is again an FC-group. The present paper extends this result to wider classes of FC-groups. Mathematics Subject Classification: 20F24; 20F14

Locally compact groups which are just not compact

A Just-Non-Compact group, or briefly a JNC group, is a Hausdorff topological group which is not a compact group but all of whose proper Hausdorff quotients are compact groups. Intuitively, it is clear that these groups are rich in compact quotients. Locally compact JNC groups are largely described in the present paper.

$MC$-hypercentral groups

This paper is devoted to the imposition of some chain conditions on groups having a generalized central series. It is also given a characterization of MC-groups with finite abelian section rank: such class of groups is a suitable enlargement of the class of FC-groups. Mathematics Subject Classification: 20F24; 20F14

On a result of L.-C. Kappe and M. Newell

There is a long line of research investigating upper central series of a group. The interest comes from the information which these series can give on the structure of a group. Baer (1952) extended the usual notion of center of a group, introducing that of p-centre, where p is a prime. Almost 40 years later, Kappe and Newell (1989) were able to embed the p-centre of a metabelian p-group in the p-th term of the upper central series. This was possible because of the growing knowledge on Engel groups of the 60s years. Here we extend the result of Kappe and Newell (1989) to wider classes of groups.

An improvement of a bound of Green

A p-group G of order pn (p prime, n ≥ 1) satisfies a classic Green's bound log p |M(G)| ≤ ½n(n - 1) on the order of the Schur multiplier M(G) of G. Ellis and Wiegold sharpened this restriction, proving that log p |M(G)| ≤ ½(d - 1)(n + m), where |G′| = pm(m ≥ 1) and d is the minimal number of generators of G. The first author has recently shown that log p |M(G)| ≤ ½(n + m - 2)(n - m - 1) + 1, improving not only Green's bound, but several other inequalities on |M(G)| in literature. Our main results deal with estimations with respect to the bound of Ellis and Wiegold.

On compact Just-Non-Lie groups

A compact group is called a compact Just-Non-Lie group or a compact JNL group if it is not a Lie group but all of its proper Hausdorff quotient groups are Lie groups. We show that a compact JNL group is profinite and a compact nilpotent JNL group is the additive group of p -adic integers for some prime. Examples show that this fails for compact pronilpotent and solvable groups.

Anti-$PC$-groups and Anti-$CC$-groups

A groupGhas Černikov classes of conjugate subgroups if the quotient groupG/coreG(NG(H))is a Černikov group for each subgroupHofG. An anti-CCgroupGis a group in which each nonfinitely generated subgroupKhas the quotient groupG/coreG(NG(K))which is a Černikov group. Analogously, a groupGhas polycyclic-by-finite classes of conjugate subgroups if the quotient groupG/coreG(NG(H))is a polycyclic-by-finite group for each subgroupHofG. An anti-PCgroupGis a group in which each nonfinitely generated subgroupKhas the quotient groupG/coreG(NG(K))which is a polycyclic-by-finite group. Anti-CCgroups and anti-PCgroups are the subject of the present article.

Elements with square roots in compact groups

The probability that a randomly chosen element has a square root is studied in [1, 2, 8] in the finite case. Here we deal with the infinite case.

A contribution to solve the problem of attributing harmonic distortion responsibility

Owing to the proliferation of loads which draw nonsinusoidal currents, the current and voltage distortion in distribution systems is ever increasing. The correct identification of the source of distortion is important for attributing the responsibility of disturbances to utility and customers. In this paper, a new classification of the possible classes of responsibility is proposed, defining some conditions, based on the improvement of the method of decomposition of the current into conforming and nonconforming components, and of the method based on the evaluation of the sign of harmonic active powers, which allows their identification.

Probability of mutually commuting n-tuples in some classes of compact groups

In finite groups the probability that two randomly chosen elements commute or randomly ordered n−tuples of elements mutually commute have recently attracted interest by many authors. There are some classical results estimating the bounds for this kind of probability so that the knowledge of the whole structure of the group can be more accurate. The same problematic has been recently extended to certain classes of infinite compact groups in [2], obtaining restrictions on the group of the inner automorphisms. Here such restrictions are improved for a wider class of infinite compact groups.

Near abelian profinite groups

Abstract A compact p-group G (p prime) is called near abelian if it contains an abelian normal subgroup A such that G/A has a dense cyclic subgroup and that every closed subgroup of A is normal in G. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular) if every pair of compact subgroups commutes setwise. We show that for p ≠ 2 a compact p-group G is near abelian if and only if it is quasihamiltonian. The case p = 2 is discussed separately.

GENERALIZED HYPERCENTERS IN INFINITE GROUPS

We consider the so-called generalized center, defined by Agrawal, in the slightly wider context of periodic groups and try to find out where additional conditions are needed for refinements. In particular we consider the final terms of the corresponding ascending sequences.