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Keywords:
free Steiner loops; Cayley graph; growth
Summary:
We study properties of Steiner loops which are of fundamental importance to develop a combinatorial theory of loops along the lines given by Combinatorial Group Theory. In a summary we describe our findings.
References:
[1] Bruck R.: Survey on Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Reihe: Gruppentheorie, Springer, Berlin-Göttingen-Heidelberg, 1958. MR 0093552
[2] Colbourn C.J., Dinitz J.H. (Eds.): Steiner Triple Systems. Section 4.5 in CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 14–15 and 70. Zbl 0984.05013
[3] Evans T.: Varieties of loops and quasigroups. in Quasigroups and Loops: Theory and Applications, ed. O. Chein, H.O. Pflugfelder, J.D.H. Smith, Heldermann, Berlin, 1990, pp. 1–26. MR 1125807 | Zbl 0721.20044
[4] Flagolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge, 2009. MR 2483235
[5] Ganter B., Pfüller U.: A remark on commutative di-associative loops. Algebra Universalis 21 (1985), 310–311. DOI 10.1007/BF01188065 | MR 0855748 | Zbl 0597.20065
[6] Grishkov A., Rasskazova D., Rasskazova M., Stuhl I.: Free Steiner triple systems and their automorphism groups. J. Algebra Appl.(to appear).
[7] Mann A.: How Groups Grow. Cambridge University Press, Cambridge, 2012. MR 2894945 | Zbl 1253.20032
[8] Markovski S., Sokolova A.: Free Steiner loops. Glasnik Matematicki 36 (2001), 85–93. MR 1842827 | Zbl 0978.08007
[9] Mwambene E.: Representing vertex-transitive graphs on groupoids. Quaest. Math. 29 (2006), 279–284. DOI 10.2989/16073600609486163 | MR 2259722 | Zbl 1107.05080
[10] Pflugfelder H.O.: Quasigroups and Loops: Introduction. Heldermann, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[11] Sabidussi G.: Vertex-transitive graphs. Monatsh. Math. 68 (1964), 426–438. DOI 10.1007/BF01304186 | MR 0175815 | Zbl 0136.44608
[12] Strambach K., Stuhl I.: Translation groups of Steiner loops. Discrete Math. 309 (2009), 4225–4227. DOI 10.1016/j.disc.2008.12.019 | MR 2519154 | Zbl 1194.20062
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