# Article

 Title: On the uniqueness of loops $M(G,2)$  (English) Author: Vojtěchovský, Petr Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 44 Issue: 4 Year: 2003 Pages: 629-635 . Category: math . Summary: 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 $i,j,k\in\{-1,\,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times\{i\})\times(G\times\{j\})$, 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 $M(G,2)$. Keyword: Moufang loops Keyword: loops $M(G, 2)$ Keyword: inverse property loops Keyword: Bol loops MSC: 20N05 idZBL: Zbl 1101.20047 idMR: MR2062879 . Date available: 2009-01-08T19:31:43Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119417 . Reference: [1] Chein O.: Moufang loops of small order.Memoirs of the American Mathematical Society, Volume 13, Issue 1, Number 197 (1978). Zbl 0378.20053, MR 0466391 Reference: [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. Zbl 0719.20036, MR 1125806 Reference: [3] Chein O., Pflugfelder H.O.: The smallest Moufang loop.Arch. Math. 22 (1971), 573-576. Zbl 0241.20061, MR 0297914 Reference: [4] Drápal A., Vojtěchovský P.: Moufang loops that share associator and three quarters of their multiplication tables.submitted. Reference: [5] Goodaire E.G., May S., Raman M.: The Moufang Loops of Order less than $64$.Nova Science Publishers, 1999. Zbl 0964.20043, MR 1689624 Reference: [6] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl 0715.20043, MR 1125767 Reference: [7] Vojtěchovský P.: The smallest Moufang loop revisited.to appear in Results Math. MR 2011917 Reference: [8] Vojtěchovský P.: Connections between codes, groups and loops.Ph.D. Thesis, Charles Univesity, 2003. .

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