Abstract
For an element \(x\) of a finite group \(G\), let Ind\(_{G}(x)\) denote the index of \(x\) in \(G\). In this note we prove that if Ind\(_{\langle a,b,x\rangle }(x)\) is a prime power for any \(a, b\in G\) of prime power order, then Ind\(_{G}(x)\) is a prime power.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aschbacher, M., Guralnick, R.: Some applications of the first cohomology group. J. Algebra 90, 446–460 (1984)
Baer, R.: Group elements of prime power index. Trans. Am. Math. Soc. 75, 20–47 (1953)
Camina, A., Camina, R.: Implications of conjugacy class size. J. Group Theory 1, 257–269 (1998)
Camina, A., Shumyatsky, P., Sica, C.: On elements of prime-power index in finite groups. J. Algebra 323, 522–525 (2010)
Gorenstein, D.: Finite Groups. Harper and Row, New York (1968)
Kazarin, L.S.: Burnside\(^{\prime }\)s \(p^{\alpha }\)-lemma. Math. Notes 48, 749–751 (1990)
Kong, Q.J.: A note on a characterisation of \(F_{2}(G)\). WSEAS Trans. Math. (in press)
Acknowledgments
The author is very grateful to Professor P. Flavell, Professor John S. Wilson and the referee for providing valuable suggestions and useful comments, which have greatly improved the final version of the paper. The paper is dedicated to Professor Péter P. Pálfy for his 60th birthday.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. S. Wilson.
The research of the author is supported by the National Natural Science Foundation of China (11301378).
Rights and permissions
About this article
Cite this article
Kong, Q. A note on elements of prime power index in finite groups. Monatsh Math 179, 577–580 (2016). https://doi.org/10.1007/s00605-015-0751-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0751-6