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CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

  • CARLO CASOLO (a1) and ELISA MARIA TOMBARI (a2)

Abstract

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

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Copyright

Corresponding author

For correspondence; e-mail: casolo@math.unifi.it

Footnotes

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The authors were partially supported by MIUR project ‘Teoria dei gruppi e applicazioni’.

Footnotes

References

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[1]Baer, R., ‘Group elements of prime power index’, Trans. Amer. Math. Soc. 75 (1953), 2047.
[2]Beltrán, A. and Felipe, M. J., ‘Variations on a theorem by Alan Camina on conjugacy class sizes’, J. Algebra 296 (2006), 253266.
[3]Beltrán, A. and Felipe, M. J., ‘Some class size conditions implying solvability of finite groups’, J. Group Theory 9 (2006), 787797.
[4]Camina, A. R., ‘Arithmetical conditions on the conjugacy class numbers of a finite group’, J. Lond. Math. Soc. (2) 2 (1972), 127132.
[5]Camina, A. R. and Camina, R. D., ‘Implications of conjugacy class size’, J. Group Theory 1 (1998), 257269.
[6]Camina, A. R. and Camina, R. D., ‘Recognizing direct products from their conjugate type vectors’, J. Algebra 234 (2000), 604608, Special issue in honor of Helmut Wielandt.
[7]Camina, A. R. and Camina, R. D., ‘Recognising nilpotent groups’, J. Algebra 300 (2006), 1624.
[8]Dolfi, S. and Lucido, M. S., ‘Finite groups in which p′-classes have q′-length’, Israel J. Math. 122 (2001), 93115.
[9]Ishikawa, K., ‘On finite p-groups which have only two conjugacy lengths’, Israel J. Math. 129 (2002), 119123.
[10]Itô, N., ‘On finite groups with given conjugate types I’, Nagoya Math. J. 6 (1953), 1728.
[11]Kurzweil, H. and Stellmacher, B., The Theory of Finite Groups (Springer, New York, 2004).
[12]Mann, A., ‘Elements of minimal breadth in finite p-groups and Lie algebras’, J. Aust. Math. Soc. 81 (2006), 209214.
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CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

  • CARLO CASOLO (a1) and ELISA MARIA TOMBARI (a2)

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