Dihedral-like constructions of automorphic loops

Mouna Aboras

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 3, page 269-284
  • ISSN: 0010-2628

Abstract

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Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if ( G , + ) is an abelian group, m ≥ 1 and α ∈ Aut ( G ) , let Dih ( m , G , α ) be defined on ℤ m × G by ( i , u ) ( j , v ) = ( i ⊕ j , ( ( - 1 ) j u + v ) α i j ) . The resulting loop is automorphic if and only if m = 2 or ( α 2 = 1 and m is even). The case m = 2 was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.

How to cite

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Aboras, Mouna. "Dihedral-like constructions of automorphic loops." Commentationes Mathematicae Universitatis Carolinae 55.3 (2014): 269-284. <http://eudml.org/doc/261864>.

@article{Aboras2014,
abstract = {Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if $(G,+)$ is an abelian group, $m\ge 1$ and $\alpha \in \operatorname\{Aut\}(G)$, let $\operatorname\{Dih\} (m,G,\alpha )$ be defined on $\mathbb \{Z\}_m\times G$ by \begin\{equation*\} (i,u)(j,v) = (i\oplus j,\,((-1)^\{j\}u + v)\alpha ^\{ij\}). \end\{equation*\} The resulting loop is automorphic if and only if $m=2$ or ($\alpha ^2=1$ and $m$ is even). The case $m=2$ was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.},
author = {Aboras, Mouna},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {dihedral automorphic loop; automorphic loop; inner mapping group; multiplication group; nucleus; commutant; center; commutator; associator subloop; derived subloop; dihedral automorphic loops; semidirect extensions; inner mapping groups; multiplication groups; nuclei; commutant; center; commutators; associator subloop; derived subloop},
language = {eng},
number = {3},
pages = {269-284},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Dihedral-like constructions of automorphic loops},
url = {http://eudml.org/doc/261864},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Aboras, Mouna
TI - Dihedral-like constructions of automorphic loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 3
SP - 269
EP - 284
AB - Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if $(G,+)$ is an abelian group, $m\ge 1$ and $\alpha \in \operatorname{Aut}(G)$, let $\operatorname{Dih} (m,G,\alpha )$ be defined on $\mathbb {Z}_m\times G$ by \begin{equation*} (i,u)(j,v) = (i\oplus j,\,((-1)^{j}u + v)\alpha ^{ij}). \end{equation*} The resulting loop is automorphic if and only if $m=2$ or ($\alpha ^2=1$ and $m$ is even). The case $m=2$ was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.
LA - eng
KW - dihedral automorphic loop; automorphic loop; inner mapping group; multiplication group; nucleus; commutant; center; commutator; associator subloop; derived subloop; dihedral automorphic loops; semidirect extensions; inner mapping groups; multiplication groups; nuclei; commutant; center; commutators; associator subloop; derived subloop
UR - http://eudml.org/doc/261864
ER -

References

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  1. Kinyon M.K., Kunen K., Phillips J.D., Vojtěchovský P., The structure of automorphic loops, to appear in Transactions of the American Mathematical Society. MR2302693
  2. Bruck R.H., A Survey of Binary Systems, Springer, 1971. Zbl0141.01401MR0093552
  3. Bruck R.H., Paige L.J., 10.2307/1969612, Ann. of Math. 2 63 (1956), 308–323. Zbl0074.01701MR0076779DOI10.2307/1969612
  4. Johnson K.W., Kinyon M.K., Nagy G.P., Vojtěchovský P., 10.1112/S1461157010000173, LMS J. Comut. Math. 14 (2011), 200–213. Zbl1225.20052MR2831230DOI10.1112/S1461157010000173
  5. Jedlička P., Kinyon M.K., Vojtěchovský P., 10.1090/S0002-9947-2010-05088-3, Trans. Amer. Math. Soc. 363 (2011), no. 1, 365–384. Zbl1215.20060MR2719686DOI10.1090/S0002-9947-2010-05088-3
  6. Jedlička P., Kinyon M.K., Vojtěchovský P., 10.1080/00927870903200877, Comm. Algebra 38 (2010), no. 9, 3243–3267. Zbl1209.20069MR2724218DOI10.1080/00927870903200877
  7. Aboras M., Dihedral-like constructions of automorphic loops, Thesis, in preparation. 

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