A class of Bol loops with a subgroup of index two

Petr Vojtěchovský

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 371-381
  • 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 . We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops M ( G , 2 ) .

How to cite

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Vojtěchovský, Petr. "A class of Bol loops with a subgroup of index two." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 371-381. <http://eudml.org/doc/249381>.

@article{Vojtěchovský2004,
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 $i, j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.},
author = {Vojtěchovský, Petr},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops; Bol loops; subloops of index two},
language = {eng},
number = {2},
pages = {371-381},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A class of Bol loops with a subgroup of index two},
url = {http://eudml.org/doc/249381},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Vojtěchovský, Petr
TI - A class of Bol loops with a subgroup of index two
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 371
EP - 381
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 $i, j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.
LA - eng
KW - Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops; Bol loops; subloops of index two
UR - http://eudml.org/doc/249381
ER -

References

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  2. Chein O., Moufang loops of small order, Memoirs of the American Mathematical Society, Volume 13, Issue 1, Number 197 (1978). Zbl0378.20053MR0466391
  3. Chein O., Pflugfelder H.O., The smallest Moufang loop, Arch. Math. 22 (1971), 573-576. (1971) Zbl0241.20061MR0297914
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  6. Drápal A., Vojtěchovský P., Moufang loops that share associator and three quarters of their multiplication tables, submitted. 
  7. Goodaire E.G., May S., Raman M., The Moufang Loops of Order less than 64 , Nova Science Publishers, 1999. Zbl0964.20043MR1689624
  8. Nagy G.P., Vojtěchovský P., LOOPS, a package for GAP 4.3. Download GAP at http://www-gap.dcs.st-and.ac.uk/ gap. Download a beta version of LOOPS at http://www.math.du.edu/loops/loops.html. 
  9. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma series in pure mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
  10. Rotman J.J., The Theory of Groups: An Introduction, Allyn and Bacon, Inc., 1965. Zbl0262.20001MR0204499
  11. Vojtěchovský P., On the uniqueness of loops M ( G , 2 ) , Comment. Math. Univ. Carolinae 44 (2003), 4 629-365. (2003) Zbl1101.20047MR2062879
  12. Vojtěchovský P., The smallest Moufang loop revisited, Results in Mathematics 44 (2003), 189-193. (2003) Zbl1050.20046MR2011917

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