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.

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 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.

Simple Moufang loops

1978 ◽  
Vol 83 (3) ◽  
pp. 377-392 ◽  
Author(s):  
Stephen Doro
Keyword(s):  
Moufang Loop ◽  
Multiplication Group ◽  
Moufang Loops

If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.

scholarly journals Some finitely based varieties of rings

1974 ◽  
Vol 17 (2) ◽  
pp. 246-255 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Nilpotent Group ◽  
Finitely Based ◽  
Finitely Generated ◽  
Nilpotent Variety ◽  
Moufang Loops ◽  
Associative Rings ◽  
Finitely Related

The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.

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 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.

BREADTH IN POLYCYCLIC GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1073-1083 ◽  
Author(s):  
AVINOAM MANN ◽  
DAN SEGAL
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Nilpotent Group ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finite Group Theory ◽  
Polycyclic Groups

The breadth of a polycyclic group is the maximum of h(G) - h(CG(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

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.

On Moufang loops that are abelian groups over the nucleus

2015 ◽  
Vol 25 (05) ◽  
pp. 889-897
Author(s):  
Piroska Csörgő ◽  
Maria L. Merlini Giuliani
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Moufang Loop ◽  
Moufang Loops ◽  
Necessary And Sufficient

We give some necessary and sufficient conditions for the equivalency of the following two properties for a Moufang loop Q: Q over the nucleus Q/N is an abelian group and Q over the center is a group. We study those properties of Moufang loops which guarantee that L(y, x) is in the automorphism group of the loop. These imply numerous statements, among them there are well-known old results.

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