On the structure of finite loop capable Abelian groups

Markku Niemenmaa

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 2, page 217-224
  • ISSN: 0010-2628

Abstract

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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 × C p × C p , where k ≥ 2 and p is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

How to cite

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Niemenmaa, Markku. "On the structure of finite loop capable Abelian groups." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 217-224. <http://eudml.org/doc/250185>.

@article{Niemenmaa2007,
abstract = {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 $k\ge 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.},
author = {Niemenmaa, Markku},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loop; group; connected transversals; loops; inner mapping groups; connected transversals},
language = {eng},
number = {2},
pages = {217-224},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the structure of finite loop capable Abelian groups},
url = {http://eudml.org/doc/250185},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Niemenmaa, Markku
TI - On the structure of finite loop capable Abelian groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 217
EP - 224
AB - 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 $k\ge 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.
LA - eng
KW - loop; group; connected transversals; loops; inner mapping groups; connected transversals
UR - http://eudml.org/doc/250185
ER -

References

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  1. Bruck R.H., Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245-354. (1946) Zbl0061.02201MR0017288
  2. Csörgö P., On connected transversals to abelian subgroups and loop theoretical consequences, Arch. Math. 86 (2006), 499-516. (2006) Zbl1113.20055MR2241599
  3. Huppert B., Endliche Gruppen I, Springer, Berlin-Heidelberg-New York, 1967. Zbl0412.20002MR0224703
  4. Kepka T., On the abelian inner permutation groups of loops, Comm. Algebra 26 (1998), 857-861. (1998) Zbl0913.20043MR1606178
  5. Kepka T., Niemenmaa M., On multiplication groups of loops, J. Algebra 135 (1990), 112-122. (1990) Zbl0706.20046MR1076080
  6. Kepka T., Niemenmaa M., On loops with cyclic inner mapping groups, Arch. Math. 60 (1993), 233-236. (1993) MR1201636
  7. Niemenmaa M., On the structure of the inner mapping groups of loops, Comm. Algebra 24 (1996), 135-142. (1996) Zbl0853.20049MR1370527
  8. Niemenmaa M., On finite loops whose inner mapping groups are abelian, Bull. Austral. Math. Soc. 65 (2002), 477-484. (2002) Zbl1012.20068MR1910500
  9. Niemenmaa M., On finite loops whose inner mapping groups are abelian II, Bull. Austral. Math. Soc. 71 (2005), 487-492. (2005) Zbl1080.20061MR2150938
  10. Niemenmaa M., Kepka T., On connected transversals to abelian subgroups, Bull. Austral. Math. Soc. 49 (1994), 121-128. (1994) Zbl0799.20020MR1262682

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