Normal Subloop Research Articles

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.

Characterization of Moufang loops whose commutant is a normal subloop

Abstract By using transversals belonging to the commutant, we characterize Moufang loops whose commutant is a normal subloop, i.e. we give necessary and sufficient conditions in the multiplication group for this purpose.

Normality, nuclear squares and Osborn identities

Let $Q$ be a loop. If $S\leq Q$ is such that $\varphi(S) \subseteq S$ for each standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that are both Buchsteiner and Osborn are characterized as loops in which each square is in the nucleus.

Steiner Loops of Affine Type

Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over {mathrm {GF}}(3). We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group A_n and we give conditions for those loops having A_n as their multiplication groups (and hence for the loops being simple).

Moufang loops with nonnormal commutative centre

AbstractWe construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).

AIM Loops and the AIM Conjecture

Summary In this article, we prove, using the Mizar [2] formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings T, L and R, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section 2, we define a set Mlt Q of multiplicative mappings of Q and cosets (mostly following Albert 1943 for cosets [1]). Next, in Section 3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set InnAut of inner mappings of Q, define the notion of an AIM loop and relate this to the conditions on T, L, and R defined by satisfies TT, etc. We prove in Theorem (67) that the nucleus of an AIM loop is normal and finally in Theorem (68) that the AIM Conjecture follows from knowing every AIM loop satisfies aa1, aa2, aa3, Ka, aK1, aK2 and aK3. The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files.

Classification of Osborn loops of order 4n

The smallest non-associative Osborn loop is of order 16. Attempts in the past to construct higher orders have been very difficult. In this paper, some examples of finite Osborn loops of order 4n, n = 4, 6, 8, 9, 12, 16 and 18 were presented. The orders of certain elements of the examples were considered. The nuclei of two of the examples were also obtained and these were used to establish the classification of these Osborn loops up to isomorphism. Moreover, the central properties of these examples were examined and were all found to be having a trivial center and no non-trivial normal subloop. Therefore, these examples of Osborn loops are simple Osborn loops.

Multiplicative Loops of Quasifields Having Complex Numbers as Kernel

We determine the multiplicative loops of locally compact connected 4-dimensional quasifields Q having the field of complex numbers as their kernel. In particular, we turn our attention to multiplicative loops which have either a normal subloop of dimension one or which contain a subgroup isomorphic to $$Spin_3({\mathbb {R}})$$ . Although the 4-dimensional semifields Q are known, their multiplicative loops have interesting Lie groups generated by left or right translations. We determine explicitly the quasifields Q which coordinatize locally compact translation planes of dimension 8 admitting an at least 16-dimensional Lie group as automorphism group.

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.

Prime Radical of Loops and Ω-Loops. I

In this paper, main properties of a commutator of two normal subloops of a loop are considered. The notion of a prime radical of loops is introduced and its characterization as a set of strongly Engel elements is given. Also an Ω-prime radical of Ω-loops is defined and its elementwise characterization is given.

Abelian Extensions and Solvable Loops

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations between nilpotence and solvability for loops and the associated multiplication groups and inner mapping groups.

On the nuclei of Moufang loops with orders coprime to six

An open problem, originally proposed by J.D. Phillips, asks if there exists an odd ordered Moufang loop that possesses a trivial nucleus. In 1968 George Glauberman proved [7] that if Q is a Moufang loop of odd order and M is any minimal normal subloop of Q whose order is coprime to its index in Q, then M is contained in the nucleus of Q. We are able to strengthen Glaubermanʼs result here by removing the coprime assumption between the order of M and its index in Q given that the loop Q has an order not divisible by three (in addition to being of odd order). Thus, a nontrivial Moufang loop having an order coprime to six certainly has a nontrivial nucleus. Concerning then the question raised by J.D. Phillips, any nontrivial Moufang loop of odd order with a trivial nucleus (should one exist) must have an order divisible by three.

CYCLIC EXTENSIONS OF MOUFANG LOOPS INDUCED BY SEMI-AUTOMORPHISMS

It is well-known that if a group G factorizes as G = NH where H ≤ G and N ⊴ G then the group structure of G is determined by the subgroups H and N, the intersection N∩H and how H acts on N with a homomorphism φ : H → Aut (N). Here, we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G = NH is a Moufang loop, N is a normal subloop, and H = 〈u〉 is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection N ∩ H, and how u permutes the elements of N as a semi-automorphism of N.

Abelian-by-Cyclic Moufang Loops

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [10] loops of this type of odd order 3q 3, with prime q ≡ 1 (mod 3), are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.

A Moufang loop's commutant

AbstractThe commutant of a loop is the set of elements which commute with all of the elements in the loop. The commutant of a Moufang loop is a subloop, but it has been an open problem to classify the Moufang loops for which the commutant is normal. It was S. Doro [3] who conjectured that a Moufang loop, under certain conditions, has a normal commutant. We settle this conjecture here by proving that the commutant of any Moufang loop is always a normal subloop.

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops. Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M 0(Q) such that Q/M 0(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C 0(Q) such that Q/C 0(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6. We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.

Power-associative, conjugacy closed loops

We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If Q is a PACC-loop with nucleus N, then Q / N is an abelian group of exponent 12; if in addition Q is finite, then | Q | is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops of order 27, four of which have the automorphic inverse property. We also study some special elements in loops, such as Moufang elements, weak inverse property (WIP) elements, and extra elements. In a CC-loop, the set of WIP and the set of extra elements are normal subloops. For each c in a PACC-loop, c 3 is WIP, c 6 is extra, and c 12 ∈ N .

Normal Subloops in the Integral Loop Ring of an RA Loop

AbstractWe show that an RA loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an RA loop can be normal in its unit loop. Over fields, this can never happen.

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeLn(G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,Ln(G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.

Bruck nets, codes, and characters of loops

Numerous computational examples suggest that if ℛk-1 ⊂ ℛ k are (k- 1)- and k-nets of order n, then rank p ℛ k - rank p ℛk-1 ≥ n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if pe ∥ n, then rank p ℛ3 ≥ 3n - 2 - e, where equality holds if and only if the loop G coordinatizing ℛ3 has a normal subloop K such that G/K is an elementary abelian group of order p e . Furthermore if n is squarefree, then rank p ℛ = 3n - 3 for every prime p ¦ n, if and only if ℛ3 is cyclic (i.e., ℛ3 is coordinated by a cyclic group of order n).