## 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.

### MSC:

 20N05 Loops, quasigroups 20N02 Sets with a single binary operation (groupoids)

Zbl 0063.00039
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