Lie rings with a~finite cyclic grading in which there are many commuting components

E. I. Khukhro

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 6 (2009), pp. 243-250.

Résumé

Let $L$ be a $(\mathbb Z/n\mathbb Z)$-graded Lie algebra (ring) with finite-dimensional (finite) zero-component of dimension $\dim L_0=r$ (of order $|L_0|=r$). If for some $m$, each grading component $L_k$ for $k\ne 0$ commutes with all but at most $m$ components, then $L$ has a soluble ideal of derived length bounded above in terms of $m$ and of codimension (index in the additive group) bounded above in terms of $n$ and $r$. If in addition $n$ is a prime, then $L$ has a nilpotent ideal of nilpotency class bounded above in terms of $m$ and of codimension (index in the additive group) bounded above in terms of $n$ and $r$. As an application, a corollary on metacyclic Frobenius groups of automorphisms is given.

Détail

Keywords: graded Lie ring, nilpotent
Mots-clés : soluble, Frobenius group, automorphism.

Bibliographie

[1] V. A. Kreknin, “Razreshimost algebr Li s regulyarnym avtomorfizmom konechnogo poryadka”, Dokl. AN SSSR, 150 (1963), 467–469 | MR | Zbl

[2] V. A. Kreknin, A. I. Kostrikin, “Algebry Li s regulyarnym avtomorfizmom”, Dokl. AN SSSR, 149 (1963), 249–251 | MR | Zbl

[3] V. D. Mazurov, “Raspoznavanie konechnykh prostykh grupp $S_4(q)$ po poryadkam ikh elementov”, Algebra i logika, 41 (2002), 166–198 | MR | Zbl

[4] N. Yu. Makarenko, “Nilpotentnyi ideal v koltsakh Li s avtomorfizmom prostogo poryadka”, Sib. matem. zh., 46 (2005), 1361–1374 | MR

[5] N. Yu. Makarenko, “Graduirovannye algebry Li s malym chislom netrivialnykh komponent”, Sib. matem. zh., 48 (2007), 116–137 | MR | Zbl

[6] E. I. Khukhro, “Konechnye $p$-gruppy, dopuskayuschie avtomorfizm poryadka $p$ s malym chislom nepodvizhnykh tochek”, Matem. zametki, 38 (1985), 652–657 | MR | Zbl

[7] E. I. Khukhro, “Gruppy i koltsa Li, dopuskayuschie pochti regulyarnyi avtomorfizm prostogo poryadka”, Matem. sb., 181 (1990), 1197–1219 | Zbl

[8] E. I. Khukhro, “Konechnye gruppy ogranichennogo ranga s pochti regulyarnym avtomorfizmom prostogo poryadka”, Sib. matem. zh., 43 (2002), 1182–1191 | MR | Zbl

[9] P. Fong, “On orders of finite groups and centralizers of $p$-elements”, Osaka J. Math., 13 (1976), 483–489 | MR | Zbl

[10] G. Higman, “Groups and rings which have automorphisms without non-trivial fixed elements”, J. London Math. Soc. (2), 32 (1957), 321–334 | DOI | MR | Zbl

[11] B. Hartley and T. Meixner, “Finite soluble groups containing an element of prime order whose centralizer is small”, Arch. Math. (Basel), 36 (1981), 211–213 | MR | Zbl

[12] E. I. Khukhro, “Graded Lie rings with many commuting components and an application to 2-Frobenius groups”, Bull. London Math. Soc., 40:5 (2008), 907–912 | DOI | MR | Zbl

[13] E. I. Khukhro and N. Yu. Makarenko, “Large characteristic subgroups satisfying multilinear commutator identities”, J. London Math. Soc., 75:3 (2007), 635–646 | DOI | MR | Zbl

[14] E. I. Khukhro, N. Yu. Makarenko, and P. Shumyatsky, “Nilpotent ideals in graded Lie algebras and almost constant-free derivations”, Commun. Algebra, 36:5 (2008), 1869–1882 | DOI | MR | Zbl

[15] N. Yu. Makarenko and E. I. Khukhro, “Almost solubility of Lie algebras with almost regular automorphisms”, J. Algebra, 277 (2004), 370–407 | MR | Zbl

[16] M. R. Pettet, “Automorphisms and Fitting factors of finite groups”, J. Algebra, 72 (1981), 404–412 | DOI | MR | Zbl

[17] A. Shalev, “Automorphisms of finite groups of bounded rank”, Israel J. Math., 82 (1993), 395–404 | DOI | MR | Zbl