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The first question is therefore open only in the infinite case. Call loop Q of Csörgõ type if it is nilpotent of class at least 3, and Inn(Q) is abelian. No loop of Csörgõ type of nilpotency class higher than 3 is known.
A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed ...
One can normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square. Then the quasigroup associated to a reduced Latin square has a (two sided) identity element (namely, the first element among the row headers). A quasigroup with a two sided identity is called a loop. Some, but not all, loops are groups.
Problems in loop theory and quasigroup theory. Add languages. ... Download QR code; Print/export Download as PDF; Printable version;
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Moufang loops are universal among inverse property loops; that is, a loop Q is a Moufang loop if and only if every loop isotope of Q has the inverse property. It follows that every loop isotope of a Moufang loop is a Moufang loop. One can use inverses to rewrite the left and right Moufang identities in a more useful form:
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In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.