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Keywords:
commutative automorphic loop; Lie algebra of characteristic $2$; nuclear extension
Summary:
In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent $2$ with trivial center.
References:
[1] Bruck R.H., Paige J.L.: Loops whose inner mappings are automorphisms. Ann. of Math. (2) 63 (1956), 308–323. DOI 10.2307/1969612 | MR 0076779 | Zbl 0074.01701
[2] Cical\`o S., de Graaf W.A., Schneider C.: LieAlgDB - a GAP package. Version 2.1, November 2010, http://www.gap-system.org/Packages/liealgdb.html
[3] Csörgö P.: The multiplication group of a finite commutative automorphic loop of order of power of an odd prime $p$ is a $p$-group. J. Algebra 350 (2012), 77–83. DOI 10.1016/j.jalgebra.2011.09.038 | MR 2859876
[4] Csörgö P.: All automorphic loops of order $p^2$ for some prime $p$ are associative. J. Algebra Appl. 12 (2013), no. 6, 1350013. DOI 10.1142/S0219498813500138 | MR 3063452 | Zbl 1275.20071
[5] de Barros D.A.S., Grishkov A., Vojtěchovský P.: Commutative automorphic loops of order $p^3$. J. Algebra Appl. 11 (2012), 1250100. DOI 10.1142/S0219498812501009 | MR 2983192
[6] Drápal A.: A class of commutative loops with metacyclic inner mapping groups. Comment. Math. Univ. Carolin. 49 (2008), no. 3, 357–382. MR 2490433
[7] Hora J., Jedlička P.: Nuclear semidirect product of commutative automorphic loops. J. Algebra Appl. 13 (2014), 1350077. DOI 10.1142/S0219498813500771 | MR 3096858 | Zbl 1292.20072
[8] Jedlička P., Kinyon M., Vojtěchovský P.: Constructions of commutative automorphic loops. Comm. Algebra 38 (2010), no. 9, 3243–3267. DOI 10.1080/00927870903200877 | MR 2724218 | Zbl 1209.20069
[9] Jedlička P., Kinyon M., Vojtěchovský P.: The structure of commutative automorphic loops. Trans. Amer. Math. Soc. 363 (2011), no. 1, 365–384. DOI 10.1090/S0002-9947-2010-05088-3 | MR 2719686 | Zbl 1215.20060
[10] Jedlička P., Kinyon M., Vojtěchovský P.: Nilpotency in automorphic loops of prime power order. J. Algebra 350 (2012), 64–76. DOI 10.1016/j.jalgebra.2011.09.034
[11] 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
[12] Kinyon M., Grishkov A., Nagy G.P.: Solvability of commutative automorphic loops. Trans. Amer. Math. Soc., to appear, arXiv:1111.7138. MR 3223359
[13] Kinyon M., Kunen K., Phillips J.D., Vojtěchovský P.: The structure of automorphic loops. in preparation, arXiv:1210.1642.
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