Affine Extensions of loops
Agota Figula, Karl Strabach

TL;DR
This paper presents a geometric method to extend loops realized in projective linear groups to affine groups, enabling new constructions in affine and unitary/orthogonal groups over specific fields.
Contribution
It introduces a simple geometric procedure for extending loops from projective linear groups to affine groups, broadening the scope of loop realizations in affine and classical groups.
Findings
Successful application to unitary groups $SU_{p_2}(n,F)$
Extension method preserves sharp transitivity in affine groups
Applicable over ordered Pythagorean $n$-real fields
Abstract
We show a simple geometric procedure for an extension of a loop realized as the image of a sharply transitive section in a subgroup of the projective linear group to a loop realized as the image of a sharply transitive section in a group of affinities of the -dimensional space over a commutative field . We desire that is a large subgroup of affine translations and that holds for the canonical homomorphism . We demonstrate that our construction successfully can be applied to sharply transitive sections in unitary and orthogonal groups of positive index over ordered pythagorean -real fields .
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Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics · Finite Group Theory Research
