Non-Commuting Graphs and Some Bounds for Commutativity Degree of Finite Moufang Loops
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Non-Commuting Graphs and Some Bounds for Commutativity Degree of Finite Moufang Loops Elhameh Rezaie1 · Karim Ahmadidelir2 · Abolfazl Tehranian1 · Hamid Rasouli1 Received: 23 April 2020 / Revised: 15 September 2020 / Accepted: 6 October 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we study some properties of the non-commuting graph Γ 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. Keywords Moufang loop · Chein loop · Non-commuting graph · Commutativity degree Mathematics Subject Classification 20N05 · 20D60 · 05C17
1 Introduction and Preliminaries Let Q be a (non-empty) set with one binary operation. Then it is a quasigroup if the equation x y = z has a unique solution in Q whenever two of the three elements
Communicated by Mohammad Shahryari.
B
Abolfazl Tehranian [email protected] Elhameh Rezaie [email protected] Karim Ahmadidelir [email protected]; [email protected] Hamid Rasouli [email protected]
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
2
Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran
123
Bulletin of the Iranian Mathematical Society
x, y, z ∈ Q are specified. A quasigroup Q is a loop if Q possesses an identity element. Moufang loops are loops in which any of the (equivalent) Moufang identities ((x y)x)z = x(y(x z)) x(y(zy)) = ((x y)z)y (x y)(zx) = x((yz)x) (x y)(zx) = (x(yz))x
(M1) (M2) (M3) (M4)
holds for every x, y, z ∈ Q. Groups are trivial examples of associative Moufang loops. Moufang loops arise naturally in algebra as the multiplicative loop of octonions, and appear in projective geometry as Moufang planes. Although Moufang loops are generally non-associative, they retain the most important properties of groups. For instance, every element x has a two-sided inverse, any two elements generate a subgroup (this property is called diassociativity), in finite Moufang loops, the order of an element divides the order of the loop. Also, as has been shown recently in [13], the order of a subloop, a non-empty subset of a loop being itself a loop under the binary operation, divides the order of the loop (Lagrange’s theorem for Moufang loops), and every finite Moufang loop of odd order is solvable. Furthermore, there are Sylow and Hall theorems for finite Moufang loops. For more details, see [6,12–14,19]. The commutator of x, y and the associator of x, y, z are defined by [x, y] = x −1 y −1 x y and [x, y, z] = ((x y)z)−1 (x(yz)), respectively, and subsequently, A(L) and L denote the subloops of L generated by the associators and commu
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