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ON VANISHING CRITERIA THAT CONTROL FINITE GROUP STRUCTURE II

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 August 2018

JULIAN BROUGH Open the ORCID record for JULIAN BROUGH [Opens in a new window]  and
QINGJUN KONG Open the ORCID record for QINGJUN KONG [Opens in a new window]
JULIAN BROUGH*
Affiliation:
Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany email [email protected]
QINGJUN KONG
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

Keywords

vanishing conjugacy classessoluble groupssupersoluble groups

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D06: Simple groups 20D08: Simple groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 251 - 257
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author gratefully acknowledges financial support by the ERC Advanced Grant 291512 and the Alexander von Humboldt Fellowship for Postdoctoral Researchers. The research of the second author is supported by the National Natural Science Foundation of China (11301378).

References

Bianchi, M., Brough, J., Camina, R. D. and Pacifici, E., ‘On vanishing class sizes in finite groups’, J. Algebra 489 (2017), 446–459.Google Scholar
Bianchi, M., Chillag, D., Lewis, M. L. and Pacifici, E., ‘Character degree graphs that are complete graphs’, Proc. Amer. Math. Soc. 135(3) (2007), 671–676 (electronic).Google Scholar
Brough, J., ‘Non-vanishing elements in finite groups’, J. Algebra 460 (2016), 387–391.Google Scholar
Brough, J., ‘On vanishing criteria that control finite group structure’, J. Algebra 458 (2016), 207–215.Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (Oxford University Press, Eynsham, 1985); with computational assistance from J. G. Thackray.Google Scholar
Dolfi, S., Pacifici, E. and Sanus, L., ‘Groups whose vanishing class sizes are not divisible by a given prime’, Arch. Math. (Basel) 94(4) (2010), 311–317.Google Scholar
Dolfi, S., Pacifici, E., Sanus, L. and Spiga, P., ‘On the orders of zeros of irreducible characters’, J. Algebra 321(1) (2009), 345–352.Google Scholar
Dolfi, S., Pacifici, E., Sanus, L. and Spiga, P., ‘On the vanishing prime graph of finite groups’, J. Lond. Math. Soc. (2) 82(1) (2010), 167–183.Google Scholar
Granville, A. and Ono, K., ‘Defect zero p-blocks for finite simple groups’, Trans. Amer. Math. Soc. 348(1) (1996), 331–347.Google Scholar
Isaacs, I. M., Character Theory of Finite Groups (Dover, New York, 1994), corrected reprint of the 1976 original, Academic Press, New York.Google Scholar
Li, S., ‘Finite groups with exactly two class lengths of elements of prime power order’, Arch. Math. (Basel) 67(2) (1996), 100–105.Google Scholar
Liu, X., Wang, Y. and Wei, H., ‘Notes on the length of conjugacy classes of finite groups’, J. Pure Appl. Algebra 196(1) (2005), 111–117.Google Scholar
Nagell, T., Introduction to Number Theory (John Wiley, New York, 1951); Almqvist and Wiksell, Stockholm, 1951.Google Scholar