Finite Moufang Loops Research Articles

Congruence Solvability in Finite Moufang Loops of Order Coprime to Three

Congruence Solvability in Finite Moufang Loops of Order Coprime to Three

A short proof for the central nilpotency of Moufang loops of prime power order

The main result is an elementary proof of the nilpotency of Moufang loops which are of prime power order. Besides basic properties of Moufang loops and Sylow theorems, the proof relies on the fact that a relative multiplication group of a (centrally nilpotent) subloop of order pk is a p-group in any finite Moufang loop Q, and that Lφ(x)−1φLxφ−1 and Rφ(x)−1φRxφ−1 are pseudoautomorphisms of Q whenever x∈Q and φ is a pseudoautomorphism of Q.

On Doro’s conjecture for finite Moufang loops

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial.

Non-Commuting Graphs and Some Bounds for Commutativity Degree of Finite Moufang Loops

In this paper, we study some properties of the non-commuting graph $$\varGamma _M$$ of a finite Moufang loop M, a graph obtained by setting all non-central elements of M as the vertex set and defining two distinct vertices to be adjacent if and only if their commutator is non-identity. In particular, Hamiltonian as well as (weak) perfectness of non-commuting graphs of Chein loops are considered. We find several upper and lower bounds for commutativity degree of some classes of finite Moufang loops by means of edge number of their non-commuting graphs and algebraic properties.

Some structural graph properties of the non-commuting graph of a class of finite Moufang loops

For any non-abelian group G , the non-commuting graph of G , Γ=Γ G , is a graph with vertex set G \ Z ( G ), where Z ( G ) is the set of elements of G that commute with every element of G and distinct non-central elements x and y of G are joined by an edge if and only if xy ≠ yx . This graph is connected for a non-abelian finite group and has received some attention in existing literature. Similarly, the non-commuting graph of a finite Moufang loop has been defined by Ahmadidelir. He has shown that this graph is connected (as for groups) and obtained some results related to the non-commuting graph of a finite non-commutative Moufang loop. In this paper, we show that the multiple complete split-like graphs are perfect (but not chordal) and deduce that the non-commuting graph of Chein loops of the form M ( D 2 n ,2) is perfect but not chordal. Then, we show that the non-commuting graph of a non-abelian group G is split if and only if the non-commuting graph of the Moufang loop M ( G ,2) is 3-split and then classify all Chein loops that their non-commuting graphs are 3-split. Precisely, we show that for a non-abelian group G , the non-commuting graph of the Moufang loop M ( G ,2), is 3-split if and only if G is isomorphic to a Frobenius group of order 2 n , n is odd, whose Frobenius kernel is abelian of order n. Finally, we calculate the energy of generalized and multiple splite-like graphs, and discuss about the energy and also the number of spanning trees in the case of the non-commuting graph of Chein loops of the form M ( D 2 n ,2).

The strong admissibility of finite Moufang loops

A loop L is strongly admissible if there exists a bijection ϕ:L→L for which the mappings g↦gϕ(g) and g↦g(gϕ(g)) are both bijections. We derive some results on the strong admissibility of finite nonassociative Moufang loops, constructed from groups using the Chein constructions.

The intersection graph of a finite Moufang loop

The intersection graph Γ SI ( G ) of a group G with identity element e is the graph whose vertex set is the set V (Γ SI ( G )) = G - e and two distinct vertices x and y are adjacent in Γ SI ( G ) if and only if |⟨ x ⟩∩⟨ y ⟩| > 1 ; where ⟨ x ⟩ is the cyclic subgroup of G generated by x . In this paper, at first we obtain some results for this graph for any Moufang loop. More specially we observe non-isomorphic finite Moufang loops may have isomorphic intersection graphs.

On the non-commuting graph in finite Moufang loops

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.

ON THE COMMUTATIVITY DEGREE IN FINITE MOUFANG LOOPS

‎The commutativity degree‎, ‎Pr(G)">Pr(G)Pr(G)‎, ‎of a finite group G">GG (i.e‎. ‎the probability that two (randomly chosen) elements of G">GGcommute with respect to its operation)) has been studied well by many authors‎. ‎It is well-known that the best upper bound for Pr(G)">Pr(G)Pr(G) is 58">5858 for a finite non-abelian group G">GG‎. ‎In this paper‎, ‎we will define the same concept for a finite non--abelian Moufang loop M">MM and try to give a best upper bound for Pr(M)">Pr(M)Pr(M)‎. ‎We will prove that for a well-known class of finite Moufang loops‎, ‎named Chein loops‎, ‎and its modifications‎, ‎this best upper bound is 2332">23322332‎. ‎So‎, ‎our conjecture is that for any finite Moufang loop M">MM‎, ‎Pr(M)≤2332">Pr(M)≤2332Pr(M)≤2332‎. ‎Also‎, ‎we will obtain some results related to the Pr(M)">Pr(M)Pr(M) and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops‎.

On Centrally and Nuclearly Nilpotent Moufang Loops

Let Q be a finite Moufang loop with nucleus N(Q) and associator subloop 𝒜(Q). First we prove if the factor loop over the nucleus Q/N has nontrivial center, then the center of Q is nontrivial too. By using this result we prove that the centrally nilpotence of Q/N(Q) implies the centrally nilpotence of 𝒜(Q), and we show that, for the centrally nilpotence of a finite Moufang loop, the centrally nilpotence of Q/N(Q) and Q/𝒜(Q) is a necessary and sufficient condition. Finally, as a corollary we give a necessary and sufficient condition for the equivalence of centrally and nuclearly nilpotence of finite Moufang loops, namely, the centrally nilpotence of Q/𝒜(Q).

Hall's Theorem for Moufang loops

A loop K is solvable if there exists a normal series 1 = K 0 ⊴ K 1 ⊴ K 2 ⊴ ⋯ ⊴ K n = K of subloops K i such that each factor K i / K i − 1 is an abelian group. Back in 1966, G. Glauberman proved [G. Glauberman, On loops of odd order II, J. Algebra 8 (1968) 393–414] that every Moufang loop of odd order is solvable. He also showed that if π is any set of primes then every Moufang loop of odd order contains a Hall π -subloop. Since then it has been an open question whether or not P. Hall's Theorem holds for all finite Moufang loops. Here we affirmatively answer this question by showing that a finite Moufang loop is solvable if and only if it contains a Hall π -subloop for any set of primes π .

The Number of Sylow p-Subloops in Finite Moufang Loops

We prove that if L is a finite Moufang loop and p is a Sylow prime for L then the number of Sylow p-subloops of L is congruent to one modulo p.

Conjugacy of Sylow 2-Subloops of the Chein Loops M 2n (G, 2)

In general, Sylow's Theorems do not hold for finite Moufang loops. It can be seen that if p is an odd prime then the Sylow p-subloops of the Chein loop M 2n (G, 2) are conjugate. Here we prove that it is also true that all the Sylow 2-subloops of M 2n (G, 2) are conjugate.

The development of Sylow p-subloops in finite Moufang loops

We show that if L is a finite Moufang loop and p is a “Sylow prime” for L then there exists a Sylow p-subloop of L. Moreover, if L 1 is a p-subloop of L then L 1 is contained in a Sylow p-subloop of L. Finally, for finite Moufang loops that don't contain Sylow p-subloops, any p-subloop can be embedded in some quasi-Sylow p-subloop.

The existence of Sylow 2-subloops in finite Moufang loops

Here we prove that if L is a finite Moufang loop then every 2-subloop of L is contained in some Sylow 2-subloop of L. Moreover, the number of Sylow 2-subloops of L is congruent to one modulo two. We will also show that if p is any prime congruent to one modulo four then there exist finite Moufang loops that do not contain any Sylow p-subloops.

Sylow's theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion for the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops.

A note on the solvability of finite Moufang loops

In this short note we show that finite Moufang loops with nilpotent inner mapping groups are solvable.

A characterization of the full wreath product

A groupoid [3, 17] is a set Q endowed with a binary product, that is, a map from Q × Q to Q. In his 1964 paper [7], Bernd Fischer studied distributive quasigroups, which by definition are groupoids Q for which right multiplication by any fixed element gives an automorphism of Q as does left multiplication. Fischer proved that the right multiplication group R(Q) of a finite distributive quasigroup Q is solvable. He did this by showing that, for a minimal counterexample, the right multiplications T = {μa : g 7→ ga | a ∈ Q } are a generating conjugacy class of involutions in R(Q) ≤ Aut(Q) with the additional property that |tr| = 3 for distinct t and r from T . He then proved that this property forces finite R(Q) to have a normal 3-group of index 2. This led Fischer to consider [8, 9, 10] the extent to which finite symmetric groups can be characterized through being generated by a conjugacy class of involutions with all products of order 1, 2, or 3—a class of 3-transpositions, since the model is the transposition (2-cycle) class of Sym(Ω), the symmetric group on the set Ω. In a landmark theorem [10], Fischer found all finite 3transposition groups with no nontrivial solvable normal subgroups, discovering three new sporadic simple groups along the way. At the same time that Fischer was considering distributive quasigroups, George Glauberman was working on certain special groupoids, called Bruck loops. Glauberman [13] proved that finite Bruck and finite Moufang loops of odd order are solvable. His approach was similar to Fischer’s. He constructed a canonical conjugacy class T of involutory loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z∗-theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime). Fischer’s and Glauberman’s work on finite quasigroups and loops had a profound effect on the theory of finite simple groups. For a normal set of involutions

Lagrange's theorem for Moufang loops

We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

Combinatorial polarization, code loops, and codes of high level

We first find the combinatorial degree of any map f : V → F, where F is a finite field and V is a finite‐dimensional vector space over F. We then simplify and generalize a certain construction, due to Chein and Goodaire, that was used in characterizing code loops as finite Moufang loops that possess at most two squares. The construction yields binary codes of high divisibility level with prescribed Hamming weights of intersections of codewords.