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Keywords:
loop; group; connected transversals
loop; group; connected transversals
Summary:
In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either $C_{p^k} \times C_{p^l}$, $k > l \geq 0$ as the Sylow $p$-subgroup for some odd prime $p$ or the group of quaternions as the Sylow $2$-subgroup may not be loop capable.
References:
[1] Baer R.: Erweiterung von Gruppen und Ihren Isomorphismen. Math. Z. 38 (1934), 375–416. DOI 10.1007/BF01170643 | MR 1545456 | Zbl 0009.01101
[2] Bruck R.: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60 (1946), 245–354. DOI 10.1090/S0002-9947-1946-0017288-3 | MR 0017288 | Zbl 0061.02201
[3] Csörgö P.: On connected transversals to abelian subgroups and loop theoretical consequences. Arch. Math. 86 (2006), 499–516. DOI 10.1007/s00013-006-1379-5 | MR 2241599
[4] Doerk K., Hawkes T.: Finite Soluble Groups. de Gruyter, Berlin, 1992. MR 1169099 | Zbl 0753.20001
[5] Drápal A.: Orbits of inner mapping groups. Monatsh. Math. 134 (2002), 191–206. DOI 10.1007/s605-002-8256-2 | MR 1883500
[6] Huppert B.: Endliche Gruppen I. Springer, Berlin-Heidelberg-New York, 1967. MR 0224703 | Zbl 0412.20002
[7] Kepka T., Niemenmaa M.: On multiplication groups of loops. J. Algebra 135 (1990), 112–122. DOI 10.1016/0021-8693(90)90152-E | MR 1076080 | Zbl 0706.20046
[8] Kepka T., Niemenmaa M.: On loops with cyclic inner mapping groups. Arch. Math. 60 (1993), 233–236. DOI 10.1007/BF01198806 | MR 1201636
[9] Mazur M.: Connected transversals to nilpotent groups. J. Group Theory 10 (2007), 195–203. DOI 10.1515/JGT.2007.015 | MR 2302614 | Zbl 1150.20010
[10] Niemenmaa M.: On finite loops whose inner mapping groups are abelian. Bull. Austral. Math. Soc. 71 (2005), 487–492. DOI 10.1017/S0004972700038491 | MR 2150938 | Zbl 1080.20061
[11] Niemenmaa M.: On the structure of finite loop capable Abelian groups. Comment. Math. Univ. Carolin. 48,2 (2007), 217–224. MR 2338090 | Zbl 1174.20345
[12] Niemenmaa M.: Finite loops with nilpotent inner mapping groups are centrally nilpotent. Bull. Aust. Math. Soc. 79 (2009), 109–114. DOI 10.1017/S0004972708001093 | MR 2486887 | Zbl 1167.20039
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