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ISSN 1088-6850(e) ISSN 0002-9947(p)

Large orbits in coprime actions of solvable groups

Author(s): Silvio Dolfi
Journal: Trans. Amer. Math. Soc. 360 (2008), 135-152.
MSC (2000): Primary 20D45; Secondary 20D20
Posted: August 20, 2007
MathSciNet review: 2341997
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Abstract | References | Similar articles | Additional information

Abstract: Let be a solvable group of automorphisms of a finite group . If $\vert G\vert$ and $\vert K\vert$ are coprime, then there exists an orbit of on of size at least $\sqrt{\vert G\vert}$ . It is also proved that in a $\pi$ -solvable group, the largest normal $\pi$ -subgroup is the intersection of at most three Hall $\pi$ -subgroups.

References:

1.: J. Dixon, The Fitting subgroup of a linear solvable group, J. Austr. Math. Soc., 7 (1967), 419-424. MR 0230814 (37:6372)
2.: S. Dolfi, Orbits of permutation groups on the power set, Arch. Math., 75 (2000), 321-327. MR 1785438 (2001g:20002)
3.: A. Espuelas, Regular orbits on symplectic modules, J. Algebra, 138 (1991), 1-12. MR 1102565 (92b:20007)
4.: S. Dolfi, Intersections of odd order Hall subgroups, Bull. London Math. Soc. 37 (2005), 61-66. MR 2105819 (2005h:20041)
[GAP]: The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.3; 2002, (http://www.gap-system.org).
5.: D. Gluck, Trivial set stabilizers in finite permutation groups, Canad. J. Math., 35 (1983), 59-76. MR 685817 (84c:20008)
6.: B. Hartley and A. Turull, On characters of coprime operator groups and the Glauberman character correspondence, J. Reine Angew. Math. 451 (1994), 175-219. MR 1277300 (95d:20010)
7.: B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin - Heidelberg - New York, 1967. MR 0224703 (37:302)
8.: B. Huppert and N. Blackburn, Finite Groups II, Springer-Verlag, Berlin - Heidelberg - New York, 1982. MR 650245 (84i:20001a)
9.: I. M. Isaacs, Large orbits in actions of nilpotent groups, Proc. Amer. Math. Soc. 127 (1999), 45-50. MR 1469413 (99b:20035)
10.: O. Manz and T. Wolf, Representations of solvable groups, Cambridge Univ. Press, Cambridge, 1993. MR 1261638 (95c:20013)
11.: H. Matsuyama, Another proof of Gluck's theorem, J. Algebra, 274 (2002), 703-706. MR 1877870 (2002j:20004)
12.: D. S. Passman, Groups with normal solvable Hall $p\, '$ -subgroups, Trans. Amer. Math. Soc. 123 (1966), 99-111. MR 0195947 (33:4143)
13.: A. Seress, The minimal base size of primitive permutation groups, J. London Math. Soc., 53 (1996), 243-255. MR 1373058 (96k:20003)
14.: M. W. Short, The primitive soluble permutation groups of degree less than , Lecture Notes in Mathematics 1519, Springer-Verlag, Berlin-Heidelberg, 1992. MR 1176516 (93g:20006)
15.: D. A. Suprunenko, Matrix groups, Translations of Mathematical Monographs 45, Amer. Math. Soc., Providence RI, 1976. MR 0390025 (52:10852)
16.: T. Wolf, Indices of centralizers for Hall-subgroups of linear groups, Illinois J. Math., 43 (1999), 324-337. MR 1703191 (2000e:20023)
17.: T. Wolf, Large orbits of supersolvable linear groups, J. Algebra 215 (1999), 235-247. MR 1684166 (2000d:20047)
18.: V. I. Zenkov, The structure of intersections of nilpotent $\pi$ -subgroups in finite $\pi$ -solvable groups, Siberian Math. J. 34 (1993), 683-687. MR 1248794 (94g:20023)

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