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On $$K$$-loops of finite order. (English) Zbl 0803.20052
This paper develops firstly the basic theory of $$K$$-loops and secondly there are given new construction methods for $$K$$-loops. For the axiomatic investigations the authors start from the quite general concept of a right loop $$(L,+)$$ (i.e., $$\forall a,b \in L\;\exists_ 1 x \in L : a+x=b$$ and $$\exists 0 \in L : \forall a \in L : 0+a=a+0=a$$). Then for $$a,b \in L$$ the maps $$a^ + : L \to L$$; $$x \to a+x$$ and $$\delta_{a,b} := ((a+b)^ +)^{-1} \circ a^ + \circ b^ +$$ are permutations. The following properties (which allow to define the various types of loops, like Bol, Bruck or $$K$$-loops) are considered: (K1$$\ell$$) $$\forall a,b \in L\;\exists_ 1 y \in L : y+a=b$$, (I) $$a+b=0 \Rightarrow b+a=0$$, (K3) $$\delta_{a,b} \in \text{Aut}(L,+)$$, (K4) $$a+b=0 \Rightarrow \delta_{a,b}=\text{id}$$, $$\text{\textbf{(K4)}}'$$ $$\delta_{a,a}=\text{id}$$, (K5) $$(-a)+(-b)=-(a+b)$$, (K6) $$\delta_{a,b}=\delta_{a,b+a}$$, (KB) (Bol-identity) $$a+(b+(a+c))=(a+(b+a))+c$$.
Besides studying the connections between these properties and the consequences by assuming some of them, the authors stress their attention on right loops with (K3). Then they add step by step further axioms and by assuming furthermore (K1$$\mathbf\ell$$), (K5) and (K6) they obtain the $$K$$-loops. To each right loop $$(L,+)$$ with (K3) and (K4) there correspond two groups, the “structure group” $$D := \langle \delta_{a,b} \mid a,b \in L\rangle$$ of $$(L,+)$$ and according to G. Kist [Result. Math. 12, 325-347 (1987; Zbl 0636.51012)] the group $$G := L \times D$$ with $$(a,\alpha) \cdot (b,\beta) := (a+\alpha(b),\delta_{a,\alpha(b)} \circ \alpha \circ \beta)$$, the so-called quasidirect product of $$L$$ and $$D$$. Therefore also the reverse problem is studied, how one can obtain loops by starting from groups (§3). This leads the authors to new construction methods (§4) for loops which they then apply (§5.6) in order to obtain new finite examples of $$K$$-loops. In particular they can show that the smallest proper $$K$$-loops have 8 elements and that there are exactly 3 non isomorphic examples. Finally they prove: If $$(L,+)$$ is a finite proper loop with (K3) and (K4) then $$| \text{Fix }\tau| \geq 2$$ for each $$\tau \in D$$.

##### MSC:
 20N05 Loops, quasigroups
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##### References:
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