Nuclear Properties of Loop Extensions

Nagy, Péter T.

doi:10.1007/s00025-019-1026-7

Cited by 3 publications

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References 33 publications

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“…Extensions of normal (sub-)loops N by (quotient) loops Q are much more relaxed than in the case of groups (cf. [2] and [8,24]), simply because for Steiner loops of affine type the associativity holds only for elements x, y and z such that x + y + z = Ω (where possibly x = y = z). We illustrate this in Example 4.1, after giving the following definition, where we brief these conditions for a Steiner loop of affine type to be such an extension (for our convenience, here we will denote the identity elements by 0).…”

Section: Theorem 8 Any Steiner Loop L S Of Affine Type Has a Subnormmentioning

confidence: 99%

Steiner Loops of Affine Type

2020

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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).

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Section: Theorem 8 Any Steiner Loop L S Of Affine Type Has a Subnormmentioning

confidence: 99%

Steiner Loops of Affine Type

2020

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Topological Loops Having Solvable Indecomposable Lie Groups as Their Multiplication Groups

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Al-Abayechi

2020

Transformation Groups

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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.

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Extensions and tangent prolongations of differentiable loops

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Nagy

2023

Journal of Algebra

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Nuclear Properties of Loop Extensions

Cited by 3 publications

References 33 publications

Steiner Loops of Affine Type

Steiner Loops of Affine Type

Topological Loops Having Solvable Indecomposable Lie Groups as Their Multiplication Groups

Extensions and tangent prolongations of differentiable loops

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