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Moufang loop
About: Moufang loop is a research topic. Over the lifetime, 217 publications have been published within this topic receiving 1802 citations.
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31 Dec 1978
TL;DR: The main result of as discussed by the authors is the determination of all non-associative Moufang loops of orders *31, including the ones referred to above, must be groups.
Abstract: The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, p , p or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2. L Introduction. In studying algebraic objects, it is frequently useful to have many examples at one's fingertips. In the case of Moufang loops that are not groups, the scarcity of manageable examples is one of the difficulties that we have encountered. It is the purpose of this paper to begin to remedy this situation by finding all Moufang loops of order < 31.0) There are 13 such loops-one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28. The order structures, nuclei and subloops of these loops are given (Tables 3, 4 and 5). All of the loops are G-loops (i.e. they are isomorphic to all of their loop isotopes) and they are solvable. Lagrange's theorem and Sylow's main theorem hold in all of them. In terms of the M^-laws of Pflugfelder [10], some of the loops are M}-loops, some are M7-loops, and some are strictly Moufang. In the course of studying these loops, we find a general method of constructing nonassociative Moufang loops as extensions of groups (see Theorem 1). We also prove that, for p and q being primes, Moufang loops of order pq or of order p" for n < 3 are groups. Presented to the Society, July 15, 1971; received by the editors November 15, 1971. AUS (MOS) subject classifications (1970). Primary 20N05.
152 citations
1 Jul 1987
TL;DR: In this paper, the Jordan-Holder theorem holds for finite simple Moufang loops, which are groups with no non-trivial proper homomorphic images, or equivalently, if it is possible to obtain a proper normal subloops.
Abstract: The purpose of this paper is to classify the finite simple Moufang loops. A Moufang loop M is a loop which satisfies the identitynote that the equivalent identities ((xy)z)y = x(y(zy)), x(y(xz)) = ((xy)x)z also hold, by [2], p. 115. The Moufang loop M is simple if it has no non-trivial proper homomorphic images, or equivalently, if it has no non-trivial proper normal subloops. For basic definitions and properties of Moufang loops, see [2] – in particular, the Jordan–Holder theorem holds for finite Moufang loops ([2], p. 67). Of course if the finite simple loop M is associative, then M is a simple group, and hence is determined by the classification of finite simple groups. In [9], Paige defines, for each finite field GF(q), a finite simple Moufang loop M(q) which is not associative – M(q) is essentially the set of units in the eight-dimensional split Cayley algebra over GF(q), modulo the centre (we shall describe M(q) in much more detail in §2).
87 citations
TL;DR: In this article, the authors consider a class of loops called Moufang loops which have a unique non-identity commutator, a unique associator, and a unique nonsmooth nonidentity square.
55 citations
50 citations
1 Mar 1955
TL;DR: In this article, a cycle of inverses is defined as a multiplicative system having a unit unit u, and a special property xr: xy *x' =y for any x and y in G. The problem which has arisen in this respect was the independence between the "inverse properties" y y'x=x and xy y' =x (equivalent to those defined in [3] ), and our property 7r.
Abstract: 1. The loop G is defined, as usual, as a multiplicative system having a unit u, and such that in the equation xy=z any two of x, y, z uniquely determine the third. Let the right inverse element of any xEG be x', so that xx'=u. We postulate a special property xr: xy *x' =y for any x and y in G. Such loops are interesting in connection with a generalization [i] of those plane webs whose study inspired the notion of the Moufang loop [2 ]. The problem which has arisen in this respect2 was the independence between the "inverse properties" y y'x=x and xy y' =x (equivalent to those defined in [3 ]) and our property 7r. Since each of the "inverse properties" implies the equality of the right and left inverses of any element, the existence of a loop with the property 7r and different right and left inverses of at least one element will be sufficient to prove the independence. In the study of our loops we use the concept of a cycle of inverses (in short: cycle), i.e. a finite sequence of elements x1, x2, . . ., xn such that x = xk+l mod n. The number n will be called the length of the cycle. A length of 1 or 2 implies identity of right and left inverses in this cycle; groups thus have only cycles of length 1 or 2. Our loops, if finite, consist only of cycles, and every element belongs exactly to one cycle. In the following we shall deal mainly with the cycles and their lengths.
35 citations
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| 2021 | 6 |
| 2020 | 4 |
| 2019 | 4 |
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| 2017 | 12 |
| 2016 | 7 |