summary:A solvable primitive group with finitely generated abelian stabilizers is finite

Ščukin, K. K.

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeK. K. ŠčukinA note on simple medial quasigroupsCommentationes Mathematicae Universitatis Carolinae, Vol. 34 (1993), No. 2, 221--222Persistent URL: http://dml.cz/dmlcz/118574Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1993Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 34,2 (1993)221–222 221A note on simple medial quasigroupsK.K. ŠčukinAbstract. A solvable primitive group with finitely generated abelian stabilizers is finite.Keywords: permutation group, primitiveClassification: 20B15In [1], J. Ježek and T. Kepka described simple medial quasigroups. Amongothers, these quasigroups turned out to be finite of prime power order. Now, usingmultiplication groups of the quasigroups (see [2]), this result can be translated intothe language of permutation groups. In the present short note we give a direct proofof the permutation group analogue. In fact, we are going to prove the following moregeneral result:Theorem. Let G be a solvable primitive permutation group on a non-empty setQ such that the stabilizers are finitely generated abelian groups. Then G is finite,Q is finite of a prime power order and the stabilizers are cyclic groups.Proof: By [3, Theorem 7, p. 37], G is the semidirect product G =M ⋋ N , whereM =M(Q,+) is the regular representation of an abelian group (Q,+) defined on Qand N is the stabilizer of the zero element 0. Moreover, since N is maximal in G, nonon-trivial proper subgroup ofM is normal in G. Further, the subring R generatedby N in the endomorphism ring of (Q,+) is a finitely generated commutative ring.Now, if q ∈ Q and f ∈ R are non-zero, then Ker (f) is a proper subgroup of (Q,+)and Ker (f) is invariant under N , which means that M(Ker (f)) is normal in Gand consequently Ker (f) = 0 and f(q) 6= 0. This implies that R(q) is a non-zerosubgroup of (Q,+) and, since it is also invariant under N , we have R(q) = Q. If0 6= p ∈ Q, then p = g(q) and q = hf(q) for suitable g, h ∈ R and hf(p) = hfg(q) =ghf(q) = g(q) = p. Thus hf = 1 and we have shown that R is a field. However,it is a well known fact that every field, finitely generated as a ring, is finite. Inparticular, R is a finite field and card (R) = card (Q) is a power of a prime number.Finally, N is a subgroup of the cyclic group R∗, and therefore N is also cyclic. References[1] Ježek J., Kepka T., Varieties of abelian quasigroups, Czech. Math. J. 27 (1977), 473–503.[2] Kepka T., Multiplication groups of some quasigroups, Colloquia Math. Soc. J. Bolyai 29(1977), 459–465.222 K.K. Ščukin[3] Supruněnko D.A., Gruppy matric, Nauka, Moskva, 1972.[4] Ščukin K.K., On simple medial quasigroups, Abstracts of talks of the international conference“Universal algebra, quasigroups and related systems”, Jadwisin, Poland, 1989, p. 32.Catedra de algebra, Universitateu de Stat diu Rep. Moldova, Strada Mateevici 60,Chişinău 14, Moldova(Received November 20, 1992)

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A note on simple medial quasigroups

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A note on simple medial quasigroups

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