It is known that Moufang loops are closely related to groups as they have many properties in common. For instance, they both have an inverse for each element and satisfy Lagrange’s theorem. In this paper, we study the properties of a class of Moufang loops which are not groups, but all their proper subloops and proper quotient loops are groups.

Let G be a finite group and C2 the cyclic group of order 2. Consider the 8 multiplicative operations (x, y) 7→ (xiyj)k, where i, j, k ∈ {−1, 1}. Define a new multiplication on G × C2 by assigning one of the above 8 multiplications to each quarter (G × {i}) × (G × {j}), for i, j ∈ C2. When G is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomo...

A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varietie...

For a given element $g$ of finite group $G$, the probablility that commutator randomly choosen pair elements in $G$ equals is relative commutativity degree $g$. In this paper we are interested studying Dihedral order $2n$ and Quaternion $2^{n}$ for any $n\geq 3$ examine infinite class Moufang Loops Chein type, $M(G,2)$.

Let L be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearly-derived subloop (normal associator subloop) L∗ of L has exponent dividing 6. It follows that Lp (the subloop of L of elements of p-power order) is associative for p > 3. Next, a loop L is said to be a small Frattini Moufang loop, or SFML, if L has a central subgroup Z of order p such that C ∼= L/Z ...

It has been shown that for distinct odd primes p1, p2, . . . , pn and q, all Moufang loops of order p1p2 · · · pnq are groups if and only if q is not congruent to 1 modulo pi for each i. In this paper, we extend that result to include Moufang loops of order p1p 2 2 · · · pnq. Mathematics Subject Classification: 20N05

Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain “group-...

C-loops are loops satisfying x(y(yz)) = ((xy)y)z. They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initiate the study of C-loops by proving: (i) Steiner loops are C-loops, (ii) C-loops are alternative, inverse property loops with squares in the nucleus, (iii) the nucleus of a C-loop is a normal subgroup, (iv) C-loops mod...