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ON THE $p$-LENGTH AND THE WIELANDT LENGTH OF A FINITE $p$-SOLUBLE GROUP

Part of: Representation theory of groups

Published online by Cambridge University Press:  07 March 2013

NING SU  and
YANMING WANG
NING SU
Affiliation:
School of Mathematics, Sun Yatsen University, Guangzhou, 510275, PR China email [email protected]
YANMING WANG*
Affiliation:
Lingnan College and School of Mathematics, Sun Yatsen University, Guangzhou, 510275, PR China
*
✉[email protected]
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Abstract

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The $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.

Keywords

p-lengthWielandt lengthnilpotent classpermutable

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 453 - 459
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Ballester-Bolinches, A. and Esteban-Romero, R., ‘Sylow permutable subnormal subgroups of finite groups’, J. Algebra 251 (2) (2002), 727–738.Google Scholar
Bauman, S., ‘$p$-normality and $p$-length of a finite group’, Math. Z. 87 (2) (1965), 345–347.Google Scholar
Bryce, R. A. and Cossey, J., ‘The Wielandt subgroup of a finite soluble group’, J. Lond. Math. Soc. 2 (2) (1989), 244.Google Scholar
Camina, A. R., ‘The Wielandt length of finite groups’, J. Algebra 15 (1) (1970), 142–148.Google Scholar
Deskins, W. E., ‘On quasinormal subgroups of finite groups’, Math. Z. 82 (2) (1963), 125–132.Google Scholar
Gorenstein, D., Finite Groups (Chelsea Publishing Company, New York, 1980).Google Scholar
Hall, P. and Higman, G., ‘On the $p$-length of $p$-soluble groups and reduction theorems for Burnside’s problem’, Proc. Lond. Math. Soc. 3 (1) (1956), 1–42.Google Scholar
Huppert, B., Endliche Gruppen, I (Springer, Berlin–Heidelberg–New York, 1967).Google Scholar
Ormerod, E. A., ‘On the Wielandt length of metabelian $p$-groups’, Arch. Math. 57 (3) (1991), 212–215.CrossRefGoogle Scholar
Qiao, S., Qian, G. and Wang, Y., ‘Influence of permutizers of subgroups on the structure of finite groups’, J. Pure Appl. Algebra 212 (10) (2008), 2307–2313.CrossRefGoogle Scholar
Schmidt, R., Subgroup Lattices of Groups, Vol. 14 (Walter de Gruyter, Berlin, 1994).Google Scholar
Wielandt, H., ‘Über den Normalisator der subnormalen Untergruppen’, Math. Z. 69 (1) (1958), 463–465.CrossRefGoogle Scholar