On the uniqueness of loops M ( G , 2 )

Petr Vojtěchovský

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 629-635
  • ISSN: 0010-2628

Abstract

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Let G be a finite group and C 2 the cyclic group of order 2. Consider the 8 multiplicative operations ( x , y ) ↦ ( x i y j ) k , where i , j , k ∈ { - 1 , 1 } . Define a new multiplication on G × C 2 by assigning one of the above 8 multiplications to each quarter ( G × { i } ) × ( G × { j } ) , for i , j ∈ C 2 . If the resulting quasigroup is a Bol loop, it is Moufang. When G is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops M ( G , 2 ) .

How to cite

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Vojtěchovský, Petr. "On the uniqueness of loops $M(G,2)$." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 629-635. <http://eudml.org/doc/249159>.

@article{Vojtěchovský2003,
abstract = {Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i,j,k\in \lbrace -1,\,1\rbrace $. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \lbrace i\rbrace )\times (G\times \lbrace j\rbrace )$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.},
author = {Vojtěchovský, Petr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops; Moufang loops; loops ; inverse property loops; Bol loops; multiplication tables},
language = {eng},
number = {4},
pages = {629-635},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the uniqueness of loops $M(G,2)$},
url = {http://eudml.org/doc/249159},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Vojtěchovský, Petr
TI - On the uniqueness of loops $M(G,2)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 629
EP - 635
AB - Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i,j,k\in \lbrace -1,\,1\rbrace $. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \lbrace i\rbrace )\times (G\times \lbrace j\rbrace )$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.
LA - eng
KW - Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops; Moufang loops; loops ; inverse property loops; Bol loops; multiplication tables
UR - http://eudml.org/doc/249159
ER -

References

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  1. Chein O., Moufang loops of small order, Memoirs of the American Mathematical Society, Volume 13, Issue 1, Number 197 (1978). Zbl0378.20053MR0466391
  2. Chein O., Pflugfelder H.O., Smith J.D.H., Quasigroups and Loops: Theory and Applications, Sigma Series in Pure Mathematics 8, Heldermann Verlag, Berlin, 1990. Zbl0719.20036MR1125806
  3. Chein O., Pflugfelder H.O., The smallest Moufang loop, Arch. Math. 22 (1971), 573-576. (1971) Zbl0241.20061MR0297914
  4. Drápal A., Vojtěchovský P., Moufang loops that share associator and three quarters of their multiplication tables, submitted. 
  5. Goodaire E.G., May S., Raman M., The Moufang Loops of Order less than 64 , Nova Science Publishers, 1999. Zbl0964.20043MR1689624
  6. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
  7. Vojtěchovský P., The smallest Moufang loop revisited, to appear in Results Math. MR2011917
  8. Vojtěchovský P., Connections between codes, groups and loops, Ph.D. Thesis, Charles Univesity, 2003. 

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