Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
metabelian groups; automorphic loops; Bruck loops; Moufang loops
metabelian groups; automorphic loops; Bruck loops; Moufang loops
Summary:
Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in ``Engelsche elemente noetherscher gruppen'' (1957) by R. Baer. We investigate some general properties and applications of ``$\circ$'' and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In ``A class of loops categorically isomorphic to Bruck loops of odd order'' (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ)$ is automorphic.
References:
[1] Baer R.: Engelsche Elemente Noetherscher Gruppen. Math. Ann. 133 (1957), 256–270 (German). DOI 10.1007/BF02547953 | MR 0086815
[2] Bender H.: Über den größten $p'$-Normalteiler in $p$-auflösbaren Gruppen. Arch. Math. (Basel) 18 (1967), 15–16 (German). DOI 10.1007/BF01899467 | MR 0213439
[3] Bruck R. H.: A Survey of Binary Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Reihe: Gruppentheorie, Springer, Berlin, 1958. MR 0093552 | Zbl 0141.01401
[4] HASH(0x23f7c08): GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. https://www.gap-system.org, 2008.
[5] Glauberman G.: On loops of odd order. J. Algebra 1 (1964), 374–396. DOI 10.1016/0021-8693(64)90017-1 | MR 0175991 | Zbl 0155.03901
[6] Greer M.: A class of loops categorically isomorphic to Bruck loops of odd order. Comm. Algebra 42 (2014), no. 8, 3682–3697. MR 3196069
[7] Isaacs I. M.: Finite Group Theory. Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, 2008. DOI 10.1090/gsm/092 | MR 2426855
[8] Jedlička P., Kinyon M., Vojtěchovský P.: The structure of commutative automorphic loops. Trans. Amer. Math. Soc. 363 (2011), no. 1, 365–384. DOI 10.1090/S0002-9947-2010-05088-3 | MR 2719686 | Zbl 1215.20060
[9] Kinyon M. K., Nagy G. P., Vojtěchovský P.: Bol loops and Bruck loops of order $pq$. J. Algebra 473 (2017), 481–512. DOI 10.1016/j.jalgebra.2016.11.023 | MR 3591160
[10] Kinyon M. K., Vojtěchovský P.: Primary decompositions in varieties of commutative diassociative loops. Comm. Algebra 37 (2009), no. 4, 1428–1444. DOI 10.1080/00927870802278917 | MR 2510992
[11] McCune W. W.: Prover9, Mace4. https://www.cs.unm.edu/̴ mccune/prover9/ , 2009.
[12] Nagy G. P., Vojtěchovský P.: Computing with small quasigroups and loops. Quasigroups Related Systems 15 (2007), no. 1, 77–94. MR 2379126
[13] Stuhl I., Vojtěchovský P.: Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order. Nonassociative Mathematics and Its Applications, Contemp. Math., 721, Amer. Math. Soc., Providence, 2019, pages 261–276. DOI 10.1090/conm/721/14510 | MR 3898514
[14] Pflugfelder H. O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[15] Thompson J. G.: Fixed points of $p$-groups acting on $p$-groups. Math. Z. 86 (1964), 12–13. DOI 10.1007/BF01111272 | MR 0168653
Search
Partner of
