On finite loops and their inner mapping groups

Markku Niemenmaa

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 341-347
  • ISSN: 0010-2628

Abstract

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In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.

How to cite

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Niemenmaa, Markku. "On finite loops and their inner mapping groups." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 341-347. <http://eudml.org/doc/249349>.

@article{Niemenmaa2004,
abstract = {In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.},
author = {Niemenmaa, Markku},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loop; group; connected transversals; finite loops; inner mapping groups; connected transversals},
language = {eng},
number = {2},
pages = {341-347},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On finite loops and their inner mapping groups},
url = {http://eudml.org/doc/249349},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Niemenmaa, Markku
TI - On finite loops and their inner mapping groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 341
EP - 347
AB - In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.
LA - eng
KW - loop; group; connected transversals; finite loops; inner mapping groups; connected transversals
UR - http://eudml.org/doc/249349
ER -

References

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  1. Baer R., Erweiterung von Gruppen und ihren Isomorphismen, Math. Z. 38 (1934), 375-416. (1934) Zbl0009.01101MR1545456
  2. Csörgö P., Myllylä K., Niemenmaa M., On connected transversals to dihedral subgroups of order 2 p n , Algebra Colloquium 7:1 (2000), 105-112. (2000) MR1810601
  3. Drápal A., Orbits of inner mapping groups, Monats. Math. 134 (2002), 191-206. (2002) Zbl1005.20051MR1883500
  4. Huppert B., Endliche Gruppen I, Springer-Verlag, 1967. Zbl0412.20002MR0224703
  5. Kepka T., On the abelian inner permutation groups of loops, Comm. Algebra 26 (1998), 857-861. (1998) Zbl0913.20043MR1606178
  6. Kepka T., Niemenmaa M., On loops with cyclic inner mapping groups, Arch. Math. 60 (1993), 233-236. (1993) MR1201636
  7. Myllylä K., On the solvability of groups and loops, Acta Universitatis Ouluensis, Series A, 396 (2002). 
  8. Niemenmaa M., On loops which have dihedral 2 -groups as inner mapping groups, Bull. Austral. Math. Soc. 52 (1995), 153-160. (1995) Zbl0838.20080MR1344268
  9. Niemenmaa M., On the structure of the inner mapping groups of loops, Comm. Algebra 24 (1996), 135-142. (1996) Zbl0853.20049MR1370527
  10. Niemenmaa M., On finite loops whose inner mapping groups are abelian, Bull. Austral. Math. Soc. 65 (2002), 477-484. (2002) Zbl1012.20068MR1910500
  11. Niemenmaa M., Finite loops with dihedral inner mapping groups are solvable, to appear in J. Algebra. Zbl1047.20051MR2032461
  12. Niemenmaa M., Kepka T., On multiplication groups of loops, J. Algebra 135 (1990), 112-122. (1990) Zbl0706.20046MR1076080
  13. Niemenmaa M., Kepka T., On connected transversals to abelian subgroups, Bull. Austral. Math. Soc. 49 (1994), 121-128. (1994) Zbl0799.20020MR1262682
  14. Vesanen A., On solvable loops and groups, J. Algebra 180 (1996), 862-876. (1996) MR1379214

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