Groupes d'Automorphismes des Boucles de Moufang Commutatives

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.

A dessin is a cellular embedding of a connected bipartite graph into an orientable closed surface with a fixed colouring of vertices and prescribed global orientation. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the set of edges, and a regular dessin is symmetric if it admits an external symmetry transposing the vertex colours. The symmetric dessins whose automorphism groups are nilpotent of class two are classified.

By a nilpotent map we mean an orientably regular map whose orientation preserving automorphism group is nilpotent. The nilpotent maps are concluded to the maps whose automorphism group is a $2$-group and a complete classification of nilpotent maps of (nilpotency) class $2$ is given by Malnič et al. in [European J. Combin. 33 (2012), 1974-1986]. It is proved by Conder et al. in [J. Algebraic Combin. 44 (2016), 863-874] that given the class, there are finitely many simple nilpotent maps. However, for the nilpotent maps with multiple edges and given class, since its automorphism group may be infinitely big, it is impossible to list it by a computer. Therefore, to classify the nilpotent maps with small class $c$ is necessary and interesting. In this paper, the nilpotent maps of class $4$ will be determined.

Moufang loops with commuting inner mappings

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Let be a group of prime exponent . In this paper we prove that is nilpotent of class at most 2 if and only if one of the following properties is true: is also the support of a commutative group such that and have the same cyclic cosets [cosets of order ]. the operation defined on by putting , gives a structure of commutative group.\end

Free commutative moufang loops and anticommutative graded rings

In this paper, we find an exact sequence involving the automorphism group of a self-dual type-A complex lattice (including the Niemeier lattice A12 2 and the complex Leech lattice) and the automorphism of a certain group central extension of the lattice. Such an exact sequence is important for the study of finite groups which have a moonshine representation analogous to that of the Monster

If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

We deal with first-order definability in the substructure ordering \(({\mathscr{D}}; \sqsubseteq )\) of finite directed graphs. In two papers, the author has already investigated the first-order language of the embeddability ordering \(({\mathscr{D}}; \leq )\). The latter has turned out to be quite strong, e.g., it has been shown that, modulo edge-reversing (on the whole graphs), it can express the full second-order language of directed graphs. Now we show that, with finitely many directed graphs added as constants, the first order language of \(({\mathscr{D}}; \sqsubseteq )\) can express that of \(({\mathscr{D}}; \leq )\). The limits of the expressive power of such languages are intimately related to the automorphism groups of the orderings. Previously, analogue investigations have found the concerning automorphism groups to be quite trivial, e.g., the automorphism group of \(({\mathscr{D}}; \leq )\) is isomorphic to \(\mathbb {Z}_{2}\). Here, unprecedentedly, this is not the case. Even though we conjecture that the automorphism group is isomorphic to \((\mathbb {Z}_{2}^{4} \times S_{4})\rtimes _{\alpha } \mathbb {Z}_{2}\), with a particular α in the semidirect product, we only prove it is finite.

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups.

In this paper, we give a generalization of the concept of group commutativity. A group [Formula: see text] will be defined as [Formula: see text]-abelian (or [Formula: see text]-commutative), where [Formula: see text] is the alternating group of degree [Formula: see text], if [Formula: see text], for every [Formula: see text], and every pairwise non-commuting elements [Formula: see text]. We prove that [Formula: see text]-commutativity implies [Formula: see text]-commutativity for every [Formula: see text]. For [Formula: see text]-groups, we prove that the converse holds. We provide some sufficient condition for [Formula: see text]-commutativity, and then we prove that Hamiltonian groups and minimal non-abelian [Formula: see text]-groups are [Formula: see text]-abelian. For finite groups, we prove that a non-abelian and [Formula: see text]-abelian group is nilpotent of class two and has even order. A partial description of such groups is given.

We have classified by computer the projectively distinct complete (k, 3)‐arcs in PG(2, q), q≤13. The algorithm used is an application of isomorph‐free backtracking using canonical augmentation, an adaptation of our earlier algorithms for the generation of (k, 2)‐arcs. We describe those parts of the algorithms which are specific to the particular problem of (k, 3)‐arcs. For each of these arcs we have also determined the automorphism group. The results are summarized in tables where the arcs are listed according to size and automorphism group. For the arcs with the larger automorphism groups, explicit descriptions are given. Part of the computer results can be generalized to other values of q: we describe constructions of arcs having S4 as a group of automorphisms, arcs containing the union of three “half conics” and arcs constructed from parts of cubic curves. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:89‐111, 2012