Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T02:36:18.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime powers as conjugacy class lengths of π-elements

Published online by Cambridge University Press:  17 April 2009

Antonio Beltrán  and
María José Felipe
Antonio Beltrán
Affiliation:
Departamento de Matemáticas, Universidad Jaume, I 12071 CastellóSpain e-mail: [email protected]
María José Felipe
Affiliation:
Departamento de Matemática Aplicada, Universided Politécnica de Valencia, 46022 ValenciaSpain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite group and π an arbitrary set of primes. We investigate the structure of G when the lengths of the conjugacy classes of its π-elements are prime powers. Under this condition, we show that such lengths are either powers of just one prime or exactly {1,qa, rb}, with q and r two distinct primes lying in π and a, b > 0. In the first case, we obtain certain properties of the normal structure of G, and in the second one, we provide a characterisation of the structure of G.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 317 - 325
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Baer, R., ‘Group elements of prime power index’, Trans. Amer. Math. Soc. 75 (1953), 20–47.CrossRefGoogle Scholar
[2]Beltrán, A. and Felipe, M.J., ‘On the diameter of a p-regular conjugacy class graph of finite groups’, Comm. Algebra 30 (2002), 5861–5873.CrossRefGoogle Scholar
[3]Beltrán, A. and Felipe, M.J., ‘Finite groups with a disconnected p-regular conjugacy class graph’, Comm. Algebra (to appear).Google Scholar
[4]Beltrán, A. and Felipe, M.J., ‘Certain relations between p-regular class sizes and the p-structure of p-solvable groups’, J. Austral. Math. Soc. (to appear).Google Scholar
[5]Bertram, E.A., Herzog, M. and Mann, A., ‘On a graph related to conjugacy classes of groups’, Bull. Lond. Math. Soc. 22 (1990), 569–575.CrossRefGoogle Scholar
[6]Bianchi, M., Chillag, D., Mauri, A.G., Herzog, M. and Scoppola, C.M., ‘Applications of a graph related to conjugacy classes in finite groups’, Arch. Math. 58 (1992), 126–132.CrossRefGoogle Scholar
[7]Camina, A.R. and Camina, R.D., ‘Implications of conjugacy class size’, J. Group Theory 1 (1998), 257–269.CrossRefGoogle Scholar
[8]Chillag, D. and Herzog, M., ‘On the lengths of the conjugacy classes of finite groups’, J. Algebra 131 (1990), 110–125.CrossRefGoogle Scholar
[9]Huppert, B., Character theory of finite groups, De Gruyter Expositions in Mathematics 25 (Walter de Gruyter & Co, Berlin, 1998).CrossRefGoogle Scholar
[10]Kazarin, S.L., ‘Burnside's p α-lemma’, (in Russian), Mat. Zametki 48 (1990), 45–48, 158. English translation, Math Notes 48 (1990), 749–751.Google Scholar
[11]Lu, Z. and Zhang, J., ‘On the diameter of a graph related to p-regular conjugacy classes of finite groups’, J. Algebra 231 (2000), 705–712.CrossRefGoogle Scholar