Topological Loops with Decomposable Solvable Multiplication Groups

In this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.

Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop

TOPOLOGICAL LOOPS HAVING SOLVABLE INDECOMPOSABLE LIE GROUPS AS THEIR MULTIPLICATION GROUPS

We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.

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 $${\mathrm {GF}}(3)$$ 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 $$A_n$$ A n and we give conditions for those loops having $$A_n$$ A n as their multiplication groups (and hence for the loops being simple).

On torsion-free nilpotent loops

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY–DICKSON LOOPS

The Cayley–Dickson loop Qn is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley–Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We establish that the inner mapping group Inn (Qn) is an elementary abelian 2-group of order 22n-2 and describe the multiplication group Mlt (Qn) as a semidirect product of Inn (Qn) × ℤ2 and an elementary abelian 2-group of order 2n. We prove that one-sided inner mapping groups Inn l(Qn) and Inn r(Qn) are equal, elementary abelian 2-groups of order 22n-1-1. We establish that one-sided multiplication groups Mlt l(Qn) and Mlt r(Qn) are isomorphic, and show that Mlt l(Qn) is a semidirect product of Inn l(Qn) × ℤ2 and an elementary abelian 2-group of order 2n.