A second grammar of associators

1978 ◽  
Vol 84 (3) ◽  
pp. 405-416 ◽  
Author(s):  
J. D. H. Smith
Keyword(s):  
Finite Number ◽  
Current Paper ◽  
Moufang Loop ◽  
Good Knowledge ◽  
Additional Material ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Commutative Moufang Loop ◽  
Moufang Loops

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

On the nilpotence class of commutative Moufang loops

1978 ◽  
Vol 84 (3) ◽  
pp. 387-404 ◽  
Author(s):  
J. D. H. Smith
Keyword(s):  
Group Theory ◽  
Nilpotent Group ◽  
Moufang Loop ◽  
Nilpotence Class ◽  
Multiplication Group ◽  
Commutative Moufang Loop ◽  
Moufang Loops

AbstractThe nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

On the non-commuting graph in finite Moufang loops

2018 ◽  
Vol 17 (04) ◽  
pp. 1850070
Author(s):  
Karim Ahmadidelir
Keyword(s):  
Abelian Group ◽  
Nilpotent Groups ◽  
Moufang Loop ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Commutative Moufang Loop ◽  
Commuting Graph ◽  
Moufang Loops ◽  
Central Elements ◽  
Vertex Set

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

The Number of Generators of a Linear p-Group

1972 ◽  
Vol 24 (5) ◽  
pp. 851-858 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Generating Set ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Minimal Generating Set

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

scholarly journals Solution of the Word Problem for Certain Types of Groups I

1956 ◽  
Vol 3 (1) ◽  
pp. 45-54 ◽  
Author(s):  
J. L. Britton
Keyword(s):  
Finite Number ◽  
Word Problem ◽  
Infinite Number ◽  
Finite System ◽  
Defining Relations ◽  
Number Of Generators ◽  
Infinite Word ◽  
Formed Part ◽  
Free Product Of Groups ◽  
Product Of Groups

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

scholarly journals On Semiprime Finitely Generated Right Alternative Weakly M-Rings

2013 ◽  
Vol 6 ◽  
pp. 9-12
Author(s):  
K. Jayalakshmi ◽  
C. Manjula
Keyword(s):  
Finite Number ◽  
Minimum Condition ◽  
Finitely Generated ◽  
Number Of Generators ◽  
Additional Assumptions

Right alternative rings, satisfying the weakly M-ring identity (w,xy,z) = x(w,y,z) are studied. Any one of the following additional assumptions imply associativity: semi prime and a finite number of generators: prime with char. ≠ 2 and minimum condition on either right ideals or on trivial left ideals, or simple and char. ≠ 2.

GROUPS WITH TRIALITY

2006 ◽  
Vol 05 (04) ◽  
pp. 441-463 ◽  
Author(s):  
ALEXANDER N. GRISHKOV ◽  
ANDREI V. ZAVARNITSINE
Keyword(s):  
Moufang Loop ◽  
Moufang Loops

Groups with triality, which arose in the papers of Glauberman and Doro, are naturally connected with Moufang loops. In this paper, we describe all possible, in a sense, groups with triality associated with a given Moufang loop. We also introduce several universal groups with triality and discuss their properties.

A Comment on Finite Nilpotent Groups of Deficiency Zero

1980 ◽  
Vol 23 (3) ◽  
pp. 313-316 ◽  
Author(s):  
Edmund F. Robertson
Keyword(s):  
Finite Group ◽  
Finite Number ◽  
Nilpotent Groups ◽  
Metacyclic Groups ◽  
Generators And Relations ◽  
Number Of Generators ◽  
Nilpotent Class ◽  
A Finite Group

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

A Moufang loop's commutant

2011 ◽  
Vol 152 (2) ◽  
pp. 193-206 ◽  
Author(s):  
STEPHEN M. GAGOLA
Keyword(s):  
Open Problem ◽  
Moufang Loop ◽  
Moufang Loops

AbstractThe commutant of a loop is the set of elements which commute with all of the elements in the loop. The commutant of a Moufang loop is a subloop, but it has been an open problem to classify the Moufang loops for which the commutant is normal. It was S. Doro [3] who conjectured that a Moufang loop, under certain conditions, has a normal commutant. We settle this conjecture here by proving that the commutant of any Moufang loop is always a normal subloop.

scholarly journals CYCLIC EXTENSIONS OF MOUFANG LOOPS INDUCED BY SEMI-AUTOMORPHISMS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350128 ◽  
Author(s):  
STEPHEN M. GAGOLA
Keyword(s):  
Automorphism Group ◽  
Cyclic Group ◽  
Binary Operation ◽  
Group Structure ◽  
Moufang Loop ◽  
Finite Cyclic Group ◽  
Moufang Loops

It is well-known that if a group G factorizes as G = NH where H ≤ G and N ⊴ G then the group structure of G is determined by the subgroups H and N, the intersection N∩H and how H acts on N with a homomorphism φ : H → Aut (N). Here, we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G = NH is a Moufang loop, N is a normal subloop, and H = 〈u〉 is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection N ∩ H, and how u permutes the elements of N as a semi-automorphism of N.

CLASSIFICATION OF MINIMAL NONASSOCIATIVE MOUFANG LOOPS OF ORDER pq3

2013 ◽  
Vol 23 (08) ◽  
pp. 1895-1908 ◽  
Author(s):  
WING LOON CHEE ◽  
STEPHEN M. GAGOLA ◽  
ANDREW RAJAH
Keyword(s):  
Abelian Group ◽  
Open Problem ◽  
Related Question ◽  
Moufang Loop ◽  
The Third ◽  
Odd Order ◽  
Moufang Loops ◽  
Mod P

An open problem in the theory of Moufang loops is to classify those loops which are minimally nonassociative, that is, loops which are nonassociative but where all proper subloops are associative. A related question is to classify all integers n for which a minimally nonassociative loop exists. In [Possible orders of nonassociative Moufang loops, Comment. Math. Univ. Carolin.41(2) (2000) 237–244], O. Chein and the third author showed that a minimal nonassociative Moufang loop of order 2q3can be constructed by using a non-abelian group of order q3. In [Moufang loops of odd order pq3, J. Algebra235 (2001) 66–93], the third author also proved that for odd primes p < q, a nonassociative Moufang loop of order pq3exists if and only if q ≡ 1 ( mod p). Here we complete the classification of minimally nonassociative Moufang loops of order pq3for primes p < q.

Sign in / Sign up

Export Citation Format

Share Document