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**Quasigroups. II.**
*(English)*
Zbl 0063.00042

Introduction: In the first part of this paper [Trans. Am. Math. Soc. 54, 507–519 (1943; Zbl 0063.00039)] we associated every quasigroup \(\mathfrak G\) with the transformation group \(\mathfrak G_r\) generated by the right and left multiplications of \(\mathfrak G\). We defined isotopy and showed that every quasigroup is isotopic to a loop, that is, a quasigroup \(\mathfrak G\) with identity element \(e\). We also defined the concept of normal divisor for all loops and showed that every normal divisor \(\mathfrak H\) of \(\mathfrak G\) is equal to \(e\Gamma\) where \(\Gamma\) is a normal divisor of \(\mathfrak G_r\).

The main purpose of this second part of our paper is that of presenting a proof, using the results above, of the Schreier refinement theorem and the consequent Jordan-Hölder theorem for arbitrary loops. We obtain also a number of special results, among them a construction of all loops \(\mathfrak G\) with a given normal divisor \(\mathfrak H\) and a given quotient loop \(\mathfrak G/\mathfrak H\). We use this in the construction of all loops of order six with a subloop, necessarily a normal divisor, of order three. We classify these loops into nonisotopic classes and show also that all quasigroups of order five are isotopic to one of two nonisotopic loops.

The main purpose of this second part of our paper is that of presenting a proof, using the results above, of the Schreier refinement theorem and the consequent Jordan-Hölder theorem for arbitrary loops. We obtain also a number of special results, among them a construction of all loops \(\mathfrak G\) with a given normal divisor \(\mathfrak H\) and a given quotient loop \(\mathfrak G/\mathfrak H\). We use this in the construction of all loops of order six with a subloop, necessarily a normal divisor, of order three. We classify these loops into nonisotopic classes and show also that all quasigroups of order five are isotopic to one of two nonisotopic loops.