In this paper we consider finite loops of specific order and we show that certain abelian groups are not isomorphic to inner mapping groups of these loops. By using our results we are able to construct a finite solvable group of order 120 which is not isomorphic to the multiplication group of a finite loop.
@article{119202,
author = {Markku Niemenmaa},
title = {On abelian inner mapping groups of finite loops},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {687-691},
zbl = {1051.20034},
mrnumber = {1800173},
language = {en},
url = {http://dml.mathdoc.fr/item/119202}
}
Niemenmaa, Markku. On abelian inner mapping groups of finite loops. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 687-691. http://gdmltest.u-ga.fr/item/119202/
Latin squares and their applications, Akademiai Kiado, Budapest, 1974. | MR 0351850 | Zbl 0283.05014
Alternating groups and latin squares, European J. Combin. 10 (1989), 175-180. (1989) | MR 0988511
On loops with cyclic inner mapping groups, Arch. Math. 60 (1993), 233-236. (1993) | MR 1201636
The classification of the finite simple Moufang loops, Math. Proc. Camb. Phil. Soc. 102 (1987), 33-47. (1987) | MR 0886433
On the structure of the inner mapping groups of loops, Comm. Algebra 24 (1996), 135-142. (1996) | MR 1370527 | Zbl 0853.20049
On multiplication groups of loops, J. Algebra 135 (1990), 112-122. (1990) | MR 1076080 | Zbl 0706.20046
On connected transversals to abelian subgroups, Bull. Austral. Math. Soc. 49 (1994), 121-128. (1994) | MR 1262682 | Zbl 0799.20020
An introduction to the theory of groups, Springer-Verlag, 1995. | MR 1307623 | Zbl 0810.20001
On connected transversals in $PSL(2,q)$, Ann. Acad. Sci. Fenn., Series A, I. Mathematica, Dissertationes 84, 1992. | MR 2714539 | Zbl 0744.20058
The group $PSL(2,q)$ is not the multiplication group of a loop, Comm. Algebra 22 (1994), 1177-1195. (1994) | MR 1261254