×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bol, G. Gewebe und Gruppen. Math Ann. 114 (1937), 414–431 · Zbl 0016.22603 · doi:10.1007/BF01594185
[2] Bruck, R. H.: A survey of binary systems. Springer – Verlag, Berlin 1958 · Zbl 0081.01704
[3] Bruck, R. H. and Paige, L. J.: Loops whose inner mappings are automorphisms. Annals Math. 63 (1956), 308–323 · Zbl 0074.01701 · doi:10.2307/1969612
[4] Burn, R. P. Finite Bol loops. Math. Proc. Cambridge Philos. Soc. 84 (1978), 377–385 · Zbl 0385.20043 · doi:10.1017/S0305004100055213
[5] Chein, O., Pflugfelder, H.O., Smith, J. D. H.: Quasigroups and Loops, Theory and Applications. Heldermann Verlag, Berlin 1990 · Zbl 0704.00017
[6] Glauberman, G.: On Loops of Odd Order. J. Algebra 1 (1966), 374–396 · Zbl 0123.01502 · doi:10.1016/0021-8693(64)90017-1
[7] Gräter, J.: Letter to the authors, 6 April 1993
[8] Im, B.: K-loops and their generalisations. Beiträge zur Geometrie und Algebra 23(1993), TUM-Bericht M 9312, 9–17 · Zbl 0864.20037
[9] Karzel, H.: Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191–206 · Zbl 0162.24101 · doi:10.1007/BF02993128
[10] Karzel, H.: The Lorentz group and the hyperbolic Geometry. Beiträge zur Geometrie und Algebra 24 (1993), TUM-Bericht M 9315, 10–22 · Zbl 0871.51011
[11] Karzel, H and Wefelscheid, H.: Groups with ah involutory antiautomorphism and K-loops; Application to Space – Time – World and hyperbolic geometry. Res. Math. 23 (1993), 338–354 · Zbl 0788.20034 · doi:10.1007/BF03322306
[12] Kepka, T.: A construction of Bruuck loops. Commentationes Math. Univ. Carolinae 25,4 (1984), 591–595. · Zbl 0563.20053
[13] Kerby, W.: Infinite sharply multiple transitive groups. Hamburger Mathematische Einzelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974
[14] Kerby, W. und Wefelscheid, H.: Bemerkungen über Fastbereiche und scharf 2-fach transitive Gruppen. Abh. Math. Sem. Uni. Hamburg 37 (1971), 20–29 · Zbl 0232.20004 · doi:10.1007/BF02993896
[15] Kerby, W. und Wefelscheid, H.: Über eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur. Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235 · Zbl 0258.17010 · doi:10.1007/BF02999699
[16] Kerby, W. and Wefelscheid, H.: Conditions of finiteness in sharply 2-transitive groups. Aequat. Math. 8 (1974), 169–172 · Zbl 0276.16029
[17] Kerby, W. and Wefelscheid, H.: The maximal subnearfield of a neardomain. J. Algebra 28 (1974), 319–325 · Zbl 0276.16029 · doi:10.1016/0021-8693(74)90043-X
[18] Kikkawa, M.: Geometry of homogeneous Lie loops. Hiroshima Math J. 5 (1975), 141–179 · Zbl 0304.53037
[19] Kist, G.: Theorie der verallgemeinerten kinematischen Räume. Beiträge zur Geometrie und Algebra 14, TUM-Bericht M 8611, München 1986 · Zbl 0636.51012
[20] Kolb, E. and Kreuzer, A.: Geometry of kinematic K-loops. Preprint. · Zbl 0852.20062
[21] Kreuzer, A.: Beispiele endlicher und unendlicher K-Loops. Res. Math. 23 (1993), 355–362 · Zbl 0788.20036 · doi:10.1007/BF03322307
[22] Kreuzer, A.: K-loops and Brück loops on \(\mathbb{R}\) \(\times\) \(\mathbb{R}\). J. of Geometry 47 (1993)
[23] Kreuzer, A.: Algebraische Struktur der relativistischen Geschwindigkeitsaddition. Beiträge zur Geometrie und Algebra 23 (1993), TUM-Bericht M9312, 31–44 · Zbl 0864.20042
[24] Kreuzer, A.: Construction of loops of even order. Beiträge zur Geometrie und Algebra 24 (1993), TUM-Bericht M9315, 10–22 · Zbl 0863.20034
[25] Niederreiter, H. and Robinson, K. H.: Bol loops of order pq. Math. Proc. Cambridge Philos. Soc. 89 (1981), 241–256 · Zbl 0463.20050 · doi:10.1017/S030500410005814X
[26] Robinson, D. A.: Bol-loops. Trans Amer. Math. Soc. 123 (1966), 341–354 · Zbl 0163.02001 · doi:10.1090/S0002-9947-1966-0194545-4
[27] Robinson, K., H.: A note on Bol loops of order 2nk. Aequationes Math. 22 (1981) 302–306 · Zbl 0478.20044 · doi:10.1007/BF02190186
[28] Sherma, B. L. and Solarin, A. R. T.: On the Classification of Bol loops of order 3p (p>3). Communicationes in Algebra 16(1), (1988), 37–55 · Zbl 0639.20049 · doi:10.1080/00927878808823560
[29] Ungar, A., A.: Thomas rotation and the parametrization of the Lorentz transformation group. Found. Phys. Lett. 1 (1988), 57–89 · doi:10.1007/BF00661317
[30] Ungar, A., A.: Weakly associative groups. Res. Math. 17(1990), 149–168 · Zbl 0699.20055 · doi:10.1007/BF03322638
[31] Ungar, A., A.: Group-like structure underlying the unit ball in real inner product spaces. Res. Math 18 (1990), 355–364 · Zbl 0718.20035 · doi:10.1007/BF03323180
[32] Ungar, A.A.: Several letters to the autors (1990–1993)
[33] Wähling, H.: Theorie der Fastkörper. Thales Verlag, Essen 1987
[34] Wefelscheid, H.: ZT-subgroups of sharply 3-transitive groups. Proc. Edinburgh Math. Soc. 23 (1980),9–14 · Zbl 0441.20004 · doi:10.1017/S0013091500003540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.