Search results for "Cyclic group"
showing 10 items of 20 documents
Gradings on the algebra of upper triangular matrices and their graded identities
2004
Abstract Let K be an infinite field and let UT n ( K ) denote the algebra of n × n upper triangular matrices over K . We describe all elementary gradings on this algebra. Further we describe the generators of the ideals of graded polynomial identities of UT n ( K ) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group.
Group theory aspects of chaotic strings
2014
Chaotic strings are a special type of non-hyperbolic coupled map lattices, exhibiting a rich structure of complex dynamical phenomena with a surprising correspondence to physical contents. Chaotic strings are generated by the Chebyshev maps T2() and T3(). In this paper we connect the Chebyshev maps via the Galois theory to the cyclic groups Z2 and Z3 and give some ideas how this fundamental connection might lead to the emergence of the familiar Lie group structure of particle physics and, finally, even to the emergence of space-time. The Z3-graded cubic and ternary algebras presented here have been introduced by R. Kerner in 1991 and then developed and elaborated in collaboration with many …
Local nearrings with dihedral multiplicative group
2004
AbstractA not necessarily zero-symmetric nearring R with a unit element is called local if the set of all non-invertible elements of R forms a subgroup of the additive group of R. It is proved that every local nearring whose multiplicative group is dihedral is finite and its additive group is either a 3-group of order at most 9 or a 2-group of order at most 32.
Groups with soluble minimax conjugate classes of subgroups
2008
A classical result of Neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. If $\mathfrak{X}$ is a class of groups, a group $G$ is said to have $\mathfrak{X}$-conjugate classes of subgroups if $G/core_G(N_G(H)) \in \mathfrak{X}$ for each subgroup $H$ of $G$. Here we study groups which have soluble minimax conjugate classes of subgroups, giving a description in terms of $G/Z(G)$. We also characterize $FC$-groups which have soluble minimax conjugate classes of subgroups.
Geometric limits of cyclic groups and the shape of Schottky space
2004
We give a local estimate of the shape of Schottky space at certain boundary points. This is done by studying the geometric limits of sequences of cyclic groups and applying the results to the generators of Schottky groups. We also obtain information on discrete, non-faithful representations close to the cusps.
Small $C^1$ actions of semidirect products on compact manifolds
2020
Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eige…
Some considerations on the nonabelian tensor square of crystallographic groups
2011
The nonabelian tensor square $G\otimes G$ of a polycyclic group $G$ is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of $G\otimes G$ by looking at that of $G$, on another hand, we study the nonabelian tensor product of pro--$p$--groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.
Polynomial identities for the Jordan algebra of upper triangular matrices of order 2
2012
Abstract The associative algebras U T n ( K ) of the upper triangular matrices of order n play an important role in PI theory. Recently it was suggested that the Jordan algebra U J 2 ( K ) obtained by U T 2 ( K ) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of U J 2 ( K ) . We describe a basis of the identities of U J 2 ( K ) when the field K is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on U J 2 ( K ) by the cyclic group Z 2 of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded ca…
Group Identities on Units of Group Algebras
2000
Abstract Let U be the group of units of the group algebra FG of a group G over a field F . Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years.
Abelian Sylow subgroups in a finite group, II
2015
Abstract Let p ≠ 3 , 5 be a prime. We prove that Sylow p-subgroups of a finite group G are abelian if and only if the class sizes of the p-elements of G are all coprime to p. This gives a solution to a problem posed by R. Brauer in 1956 (for p ≠ 3 , 5 ).