Moufang loops with commuting inner mappings

Moufang loops of order $2m$

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415–417] proved that a nilpotent Moufang loop is the direct product of [Formula: see text]-loops for some primes [Formula: see text], consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.

In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Dr\'{a}pal and Jedli\u{c}ka. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Dr\'{a}pal and Jedli\u{c}ka nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws $\{Q_i\}_{i=1}^{12}$ induced by our nuclear identification form four cycles in the following sequential format: $\big(Q_{4i-j}\big)_{i=1}^3,~j=0,1,2,3,$ and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities $\{Q_i^*\}_{i=1}^{12}$ induced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws $\{Q_i\}_{i=1}^{12}$ were found.

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.

Nilpotency in automorphic loops of prime power order

Groupes d'Automorphismes des Boucles de Moufang Commutatives

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$. We obtain two results bounding the exponent of $[G,\phi]$. Denote by $G_{-\phi}$ the set $\{[g,\phi]\,\vert\, g\in G\}$ and by $G_{\phi}$ the centralizer of $\phi$, that is, the subgroup of fixed points of $\phi$. The obtained results are as follows.1. Assume that the subgroup $\langle x,y\rangle$ has derived length at most $d$ and $x^e=1$ for every $x,y\in G_{-\phi}$. Suppose that $G_\phi$ is nilpotent of class $c$. Then the exponent of $[G,\phi]$ is $(c,d,e)$-bounded.2. Assume that $G_\phi$ has rank $r$ and $x^e=1$ for each $x\in G_{-\phi}$. Then the exponent of $[G,\phi]$ is $(e,r)$-bounded.

A left Bol loop is a loop satisfying x ( y ( x z ) ) = ( x ( y x ) ) z x(y(xz)) = (x(yx))z . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2 k 2k , k k odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3 3 , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop K K such that K K is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order 16 16 with a non-subloop commutant.

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.

Free commutative moufang loops and anticommutative graded rings

We prove that the maximal subloops of the simple Moufang Loop of order 1080 have orders of 120 and 108 and are unique up to isomorphism.

Some Non-split Extensions of the Orthogonal Group Ω(7, 3)

An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and (xy) -1 = x -1 y -1 holds. Let Q be a finite commutative A-loop and p a prime. The loop Q has order a power of p if and only if every element of Q has order a power of p. The loop Q decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If Q is of odd order, it is solvable. If A is a subloop of Q, then |A| divides |Q|. If p divides |Q|, then Q contains an element of order p. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.

In this chapter we treat central extensions of groups, a particular case of the construction in Sect. 1.5, and we give a complete cohomological characterization of these. Following Mihailovs (The orbit method for finite groups of nilpotency class two of odd order. Preprint: arXiv.org: math.RT/0001092) and Kokhas (J Math Sci (NY) 131(2):5508–5555, 2004) with a 2-step nilpotent group with 2-divisible center we associate a 2-step nilpotent Lie ring. This is a key construction for the definition and application of the orbit method.