The group of units as a maximal inverse subsemigroup

1982 ◽  
Vol 25 (1) ◽  
pp. 381-382
Author(s):  
Bridget B. Baird
Keyword(s):  
Group Of Units ◽  
Inverse Subsemigroup

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

scholarly journals A correspondence between a class of monoids and self-similar group actions II

2015 ◽  
Vol 25 (04) ◽  
pp. 633-668
Author(s):  
Mark V. Lawson ◽  
Alistair R. Wallis
Keyword(s):  
Group Actions ◽  
Similar Group ◽  
Universal Group ◽  
Length Functions ◽  
Group Of Units ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Self Similar ◽  
Cancellative Monoids

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

Presentation of the group of units of ZD-16

2006 ◽  
pp. 305-314
Author(s):  
Antonio Pita ◽  
Ángel del Río
Keyword(s):  
Group Of Units

scholarly journals A 5-Engel associative algebra whose group of units is not 5-Engel

2019 ◽  
Vol 519 ◽  
pp. 101-110
Author(s):  
Galina Deryabina ◽  
Alexei Krasilnikov
Keyword(s):  
Associative Algebra ◽  
Group Of Units

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

scholarly journals STRUCTURES CONCERNING GROUP OF UNITS

2017 ◽  
Vol 54 (1) ◽  
pp. 177-191
Author(s):  
Young Woo Chung ◽  
Yang Lee
Keyword(s):  
Group Of Units

scholarly journals Stratigraphy and tectonics of a tectonic window in the Magura Nappe (Świątkowa Wielka, Polish Outer Carpathians)

2011 ◽  
Vol 62 (2) ◽  
pp. 139-154 ◽  
Author(s):  
Marta Oszczypko-Clowes ◽  
Nestor Oszczypko
Keyword(s):  
Calcareous Nannoplankton ◽  
Group Of Units ◽  
Marginal Part ◽  
Tectonic Window ◽  
Outer Carpathians ◽  
Tectonic Windows

Stratigraphy and tectonics of a tectonic window in the Magura Nappe (Świątkowa Wielka, Polish Outer Carpathians)The Świątkowa Wielka Tectonic Window belongs to the Grybów Nappe of the Fore-Magura Group of units. This tectonic window is located in the marginal part of the Magura Nappe and is composed of Oligocene — Sub-Grybów Beds as well as the Grybów Marl Formation. These beds have been correlated with the Oligocene deposits of other tectonic windows of the Grybów Nappe in Poland. Our research reveals that the Krosno beds' shally facies, which occur at the western termination of the Świątkowa Wielka Tectonic Window, belong to the Dukla succession. On the basis of calcareous nannoplankton analysis, the Grybów Marl Formation as well as the Krosno Beds belong to the NP23-NP24, and NP24 Zones, respectively. The structure of the Świątkowa Wielka Tectonic Window reveals a multistage evolution of the Magura Nappe overthrust onto their foreland.

scholarly journals Rings with a cyclic group of units

1970 ◽  
Vol 16 (2) ◽  
pp. 243-251 ◽  
Author(s):  
K.R Pearson ◽  
J.E Schneider
Keyword(s):  
Cyclic Group ◽  
Group Of Units

The ideal class group and the group of units

1991 ◽  
pp. 251-261
Author(s):  
Helmut Koch
Keyword(s):  
Class Group ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Group Of Units ◽  
The Ideal
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