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# QUASIGROUPS AND LOOPS

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# QUASIGROUPS AND LOOPS

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QUASIGROUPS AND LOOPS book

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## ABSTRACT

Quasigroups may be defined combinatorially or equationally. A (combinatorial) quasigroup (Q, ·) is a set Q equipped with a binary multiplication operation

Q×Q→ Q; (x, y) 7→ xy (1.1) denoted by · or simple juxtaposition of the two arguments, in which specification of any two of x, y, z in the equation x · y = z determines the third uniquely. Note that group multiplications have this property, so that any group is a quasigroup. In particular, a quasigroup is said to be abelian if its multiplication is commutative and associative. However, quasigroup multiplications are not required to be associative. It is in this sense that quasigroups are considered to be “nonassociative groups.” Finite quasigroups are characterized in Section 1.1 as having bordered Latin

squares for their multiplication tables. The more general and precise equational definition of Section 1.2 describes quasigroups as universal algebras with operations of multiplication, left and right division. Along with homomorphisms, quasigroups may also be related by homotopies. New quasigroups, known as conjugates, are obtained by regarding the divisions as basic multiplications (Section 1.3). For example, the nonassociative operation of subtraction yields a conjugate of an abelian group. The conjugates of a given quasigroup fit together to form its semisymmetrization, so that homotopies between quasigroups correspond to homomorphisms between their semisymmetrizations (Section 1.4). The type of a quasigroup may often be augmented by an idempotent element to give a so-called “pique” or pointed idempotent quasigroup (Section 1.5). If this idempotent element acts as an identity for the multiplication, then the pique becomes a loop. Steiner triple systems are presented in Section 1.6 as an important equa-

tionally defined class of quasigroups. Section 1.7 provides a quick introduction to Moufang loops, more especially those obtained from Zorn’s vector matrices and octonions. (One of the classical motivations for studying quasigroups and loops is the need for a deeper understanding of the octonions, as the last step in the sequence of algebras that starts with the reals and leads through the complex numbers and quaternions.) Section 1.8 examines the symmetry that holds between the various conjugates of a quasigroup. This symmetry is then applied to the proof of the normal form theorem for quasigroup words in the final Section 1.9.