# Article

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Keywords:
loop; group; connected transversals
Summary:
Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups $C_p^{k}\times C_p\times C_p$, where $k\geq 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.
References:
[1] Bruck R.H.: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60 (1946), 245-354. MR 0017288 | Zbl 0061.02201
[2] Csörgö P.: On connected transversals to abelian subgroups and loop theoretical consequences. Arch. Math. 86 (2006), 499-516. MR 2241599 | Zbl 1113.20055
[3] Huppert B.: Endliche Gruppen I. Springer, Berlin-Heidelberg-New York, 1967. MR 0224703 | Zbl 0412.20002
[4] Kepka T.: On the abelian inner permutation groups of loops. Comm. Algebra 26 (1998), 857-861. MR 1606178 | Zbl 0913.20043
[5] Kepka T., Niemenmaa M.: On multiplication groups of loops. J. Algebra 135 (1990), 112-122. MR 1076080 | Zbl 0706.20046
[6] Kepka T., Niemenmaa M.: On loops with cyclic inner mapping groups. Arch. Math. 60 (1993), 233-236. MR 1201636
[7] Niemenmaa M.: On the structure of the inner mapping groups of loops. Comm. Algebra 24 (1996), 135-142. MR 1370527 | Zbl 0853.20049
[8] Niemenmaa M.: On finite loops whose inner mapping groups are abelian. Bull. Austral. Math. Soc. 65 (2002), 477-484. MR 1910500 | Zbl 1012.20068
[9] Niemenmaa M.: On finite loops whose inner mapping groups are abelian II. Bull. Austral. Math. Soc. 71 (2005), 487-492. MR 2150938 | Zbl 1080.20061
[10] Niemenmaa M., Kepka T.: On connected transversals to abelian subgroups. Bull. Austral. Math. Soc. 49 (1994), 121-128. MR 1262682 | Zbl 0799.20020

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