Previous |  Up |  Next

Article

Keywords:
finite loops; finite Bruck loops; finite Bol loops; finite $A_r$-loops; classical theorems for finite loops
Summary:
This note contains Sylow's theorem, Lagrange's theorem and Hall's theorem for finite Bruck loops. Moreover, we explore the subloop structure of finite Bruck loops.
References:
[AA43] Albert A.A.: Quasigroups I. Trans. Amer. Math. Soc. 54 (1943), 507–519. DOI 10.1090/S0002-9947-1943-0009962-7 | MR 0009962 | Zbl 0063.00039
[A86] Aschbacher M.: Finite Group Theory. Cambridge University Press, Cambridge, 1986. MR 0895134 | Zbl 0997.20001
[A05] Aschbacher M.: On Bol loops of exponent $2$. J. Algebra 288 (2005), 99–136. DOI 10.1016/j.jalgebra.2005.03.005 | MR 2138373 | Zbl 1090.20037
[AKP06] Aschbacher M., Kinyon M., Phillips J.D.: Finite Bruck loops. Trans. Amer. Math. Soc. 358 (2006), no. 7, 3061–3075. DOI 10.1090/S0002-9947-05-03778-5 | MR 2216258 | Zbl 1102.20046
[Ba39] Baer R.: Nets and groups. Trans. Amer. Math. Soc 47 (1939), 110–141. MR 0000035 | Zbl 0023.21502
[Ba45] Baer R.: The homomorphism theorems for loops. Amer. J. Math. 67 (1945), 450–460. DOI 10.2307/2371960 | MR 0012302 | Zbl 0063.00166
[Bo37] Bol G.: Gewebe und Gruppen. Math. Ann. 114 (1937), 414–431. DOI 10.1007/BF01594185 | MR 1513147 | Zbl 0016.22603
[BS10] Baumeister B., Stein A.: Self-invariant $1$-factorizations of complete graphs and finite Bol loops of exponent $2$. Beiträge Algebra Geom. 51 (2010), no. 1, 117–135. MR 2650481 | Zbl 1208.20064
[BS11] Baumeister B., Stein A.: The finite Bruck loops. J. Algebra 330 (2011), 206–220. DOI 10.1016/j.jalgebra.2010.11.017 | MR 2774625 | Zbl 1235.20059
[BSS11] Baumeister B., Stein A., Stroth G.: On Bruck loops of $2$-power exponent. J. Algebra 327 (2011), 316–336. DOI 10.1016/j.jalgebra.2010.10.033 | MR 2746041 | Zbl 1233.20059
[Br58] Bruck R.H.: A survey of binary systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Reihe: Gruppentheorie, Springer, Berlin-Göttingen-Heidelberg, 1958. MR 0093552 | Zbl 0141.01401
[Bu78] Burn R.P.: Finite Bol loops. Math. Proc. Cambridge Philos. Soc. 84 (1978), 377–385. DOI 10.1017/S0305004100055213 | MR 0492030 | Zbl 0571.20069
[Ga11] Gagola S.M., III: How and why Moufang loops behave like groups. Quasigroups Related Systems 19 (2011), 1–22. MR 2850316
[G64] Glauberman G.: On loops of odd order. J. Algebra 1 (1964), 374–396. DOI 10.1016/0021-8693(64)90017-1 | MR 0175991 | Zbl 0155.03901
[G68] Glauberman G.: On loops of odd order II. J. Algebra 8 (1968), 393–414. DOI 10.1016/0021-8693(68)90050-1 | MR 0222198 | Zbl 0155.03901
[JKV11] Jedlička P., Kinyon M.K., Vojtěchovský P.: The structure of commutative automorphic loops. Trans. Amer. Math. Soc. 363 (2011), 365–384. DOI 10.1090/S0002-9947-2010-05088-3 | MR 2719686 | Zbl 1215.20060
[JKNV11] Johnson K.W., Kinyon M.K., Nagy G.P., Vojtěchovský P.: Searching for small simple automorphic loops. LMS J. Comput. Math. 14 (2011), 200–213. DOI 10.1112/S1461157010000173 | MR 2831230 | Zbl 1225.20052
[N98] Nagy G.P.: Solvability of universal Bol $2$-loops. Comm. Algebra 26 (1998), no. 1, 549–555. DOI 10.1080/00927879808826146 | MR 1604103 | Zbl 0895.20054
[N08] Nagy G.P.: A class of simple proper Bol loops. Manuscripta Math. 127 (2008), no. 1, 81–88. DOI 10.1007/s00229-008-0188-5 | MR 2429915 | Zbl 1167.20038
[N09] Nagy G.P.: A class of finite simple Bol loops of exponent $2$. Trans. Amer. Math. Soc. 361 (2009), 5331–5343. DOI 10.1090/S0002-9947-09-04646-7 | MR 2515813
[N] Nagy G.P.: Finite simple left Bol loops. \begin{verbatim} http://www.math.u-szeged.hu/ nagyg/pub/simple_bol_loops.html \end{verbatim}.
[S44] Smiley M.F.: An application of lattice theory to quasigroups. Bull. Amer. Math. Soc. 21 (1944), 782–786. DOI 10.1090/S0002-9904-1944-08237-2 | MR 0011312 | Zbl 0063.07085
Partner of
EuDML logo