## The structure of automorphic loops

HTML articles powered by AMS MathViewer

- by Michael K. Kinyon, Kenneth Kunen, J. D. Phillips and Petr Vojtěchovský PDF
- Trans. Amer. Math. Soc.
**368**(2016), 8901-8927 Request permission

## Abstract:

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops.

We study uniquely $2$-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman’s work on uniquely $2$-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop $Q$ of odd order is solvable and contains an element of order $p$ for every prime $p$ dividing $|Q|$, and that $|S|$ divides $|Q|$ for every subloop $S$ of $Q$.

There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than $2500$. We show that if $Q$ is a finite simple nonassociative automorphic loop, then the socle of the multiplication group of $Q$ is not regular. The existence of a finite simple nonassociative automorphic loop remains open.

Let $p$ be an odd prime. Automorphic loops of order $p$ or $p^2$ are groups, but there exist nonassociative automorphic loops of order $p^3$, some with trivial nucleus (center) and of exponent $p$. We construct nonassociative “dihedral” automorphic loops of order $2n$ for every $n>2$, and show that there are precisely $p-2$ nonassociative automorphic loops of order $2p$, all of them dihedral.

## References

- A. A. Albert,
*Quasigroups. I*, Trans. Amer. Math. Soc.**54**(1943), 507–519. MR**9962**, DOI 10.1090/S0002-9947-1943-0009962-7 - Richard Hubert Bruck,
*A survey of binary systems*, Reihe: Gruppentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR**0093552**, DOI 10.1007/978-3-662-35338-7 - R. H. Bruck and Lowell J. Paige,
*Loops whose inner mappings are automorphisms*, Ann. of Math. (2)**63**(1956), 308–323. MR**76779**, DOI 10.2307/1969612 - R. P. Burn,
*Finite Bol loops*, Math. Proc. Cambridge Philos. Soc.**84**(1978), no. 3, 377–385. MR**492030**, DOI 10.1017/S0305004100055213 - Piroska Csörgő,
*Multiplication groups of commutative automorphic $p$-loops of odd order are $p$-groups*, J. Algebra**350**(2012), 77–83. MR**2859876**, DOI 10.1016/j.jalgebra.2011.09.038 - Piroska Csörgő,
*All automorphic loops of order $p^2$ for some prime $p$ are associative*, J. Algebra Appl.**12**(2013), no. 6, 1350013, 8. MR**3063452**, DOI 10.1142/S0219498813500138 - Piroska Csörgő,
*All finite automorphic loops have the elementwise Lagrange property*, Rocky Mountain J. Math.**45**(2015), no. 4, 1101–1105. MR**3418184**, DOI 10.1216/RMJ-2015-45-4-1101 - Dylene Agda Souza De Barros, Alexander Grishkov, and Petr Vojtěchovský,
*Commutative automorphic loops of order $p^3$*, J. Algebra Appl.**11**(2012), no. 5, 1250100, 15. MR**2983192**, DOI 10.1142/S0219498812501009 - John D. Dixon and Brian Mortimer,
*Permutation groups*, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR**1409812**, DOI 10.1007/978-1-4612-0731-3 - Aleš Drápal,
*A-loops close to code loops are groups*, Comment. Math. Univ. Carolin.**41**(2000), no. 2, 245–249. Loops’99 (Prague). MR**1780868** - Tuval Foguel, Michael K. Kinyon, and J. D. Phillips,
*On twisted subgroups and Bol loops of odd order*, Rocky Mountain J. Math.**36**(2006), no. 1, 183–212. MR**2228190**, DOI 10.1216/rmjm/1181069494 - The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10; 2007. http://www.gap-system.org
- George Glauberman,
*On loops of odd order*, J. Algebra**1**(1964), 374–396. MR**175991**, DOI 10.1016/0021-8693(64)90017-1 - George Glauberman,
*On loops of odd order. II*, J. Algebra**8**(1968), 393–414. MR**222198**, DOI 10.1016/0021-8693(68)90050-1 - Alexander Grishkov, Michael Kinyon, and Gábor P. Nagy,
*Solvability of commutative automorphic loops*, Proc. Amer. Math. Soc.**142**(2014), no. 9, 3029–3037. MR**3223359**, DOI 10.1090/S0002-9939-2014-12053-3 - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR**499562** - Přemysl Jedlička, Michael Kinyon, and Petr Vojtěchovský,
*The structure of commutative automorphic loops*, Trans. Amer. Math. Soc.**363**(2011), no. 1, 365–384. MR**2719686**, DOI 10.1090/S0002-9947-2010-05088-3 - Přemysl Jedlička, Michael K. Kinyon, and Petr Vojtěchovský,
*Constructions of commutative automorphic loops*, Comm. Algebra**38**(2010), no. 9, 3243–3267. MR**2724218**, DOI 10.1080/00927870903200877 - Přemysl Jedlička, Michael Kinyon, and Petr Vojtěchovský,
*Nilpotency in automorphic loops of prime power order*, J. Algebra**350**(2012), 64–76. MR**2859875**, DOI 10.1016/j.jalgebra.2011.09.034 - Kenneth W. Johnson, Michael K. Kinyon, Gábor P. Nagy, and Petr Vojtěchovský,
*Searching for small simple automorphic loops*, LMS J. Comput. Math.**14**(2011), 200–213. MR**2831230**, DOI 10.1112/S1461157010000173 - Michael K. Kinyon, Kenneth Kunen, and J. D. Phillips,
*Every diassociative $A$-loop is Moufang*, Proc. Amer. Math. Soc.**130**(2002), no. 3, 619–624. MR**1866009**, DOI 10.1090/S0002-9939-01-06090-7 - W. W. McCune,
*Prover9 and Mace4*, version 2009-11A. http://www.cs.unm.edu/~mccune/prover9/ - G. P. Nagy and P. Vojtěchovský,
*LOOPS: Computing with quasigroups and loops in GAP*, version 2.0.0, computational package for GAP; http://www.math.du.edu/loops - J. Marshall Osborn,
*A theorem on $A$-loops*, Proc. Amer. Math. Soc.**9**(1958), 347–349. MR**93555**, DOI 10.1090/S0002-9939-1958-0093555-6 - Hala O. Pflugfelder,
*Quasigroups and loops: introduction*, Sigma Series in Pure Mathematics, vol. 7, Heldermann Verlag, Berlin, 1990. MR**1125767** - D. A. Robinson,
*Bol quasigroups*, Publ. Math. Debrecen**19**(1972), 151–153 (1973). MR**325829** - C. R. B. Wright,
*Nilpotency conditions for finite loops*, Illinois J. Math.**9**(1965), 399–409. MR**181691** - C. R. B. Wright,
*On the multiplication group of a loop*, Illinois J. Math.**13**(1969), 660–673. MR**248270**

## Additional Information

**Michael K. Kinyon**- Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
- MR Author ID: 267243
- ORCID: 0000-0002-5227-8632
- Email: mkinyon@du.edu
**Kenneth Kunen**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 57306
- MR Author ID: 107920
- Email: kunen@math.wisc.edu
**J. D. Phillips**- Affiliation: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, Michigan 49855
- MR Author ID: 322053
- Email: jophilli@nmu.edu
**Petr Vojtěchovský**- Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
- MR Author ID: 650320
- Email: petr@math.du.edu
- Received by editor(s): October 4, 2012
- Received by editor(s) in revised form: November 26, 2014
- Published electronically: March 21, 2016
- Additional Notes: The fourth author was partially supported by Simons Foundation Collaboration Grant 210176.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 8901-8927 - MSC (2010): Primary 20N05
- DOI: https://doi.org/10.1090/tran/6622
- MathSciNet review: 3551593