This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power g n. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.
@article{bwmeta1.element.doi-10_2478_s11533-013-0287-8,
author = {P\'al Heged\H us},
title = {Groups where each element is conjugate to its certain power},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1742-1749},
zbl = {1291.20015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0287-8}
}
Pál Hegedűs. Groups where each element is conjugate to its certain power. Open Mathematics, Tome 11 (2013) pp. 1742-1749. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0287-8/
[1] Farias e Soares E., Big primes and character values for solvable groups, J. Algebra, 1986, 100(2), 305–324 http://dx.doi.org/10.1016/0021-8693(86)90079-7[Crossref]
[2] Gow R., Groups whose characters are rational-valued, J. Algebra, 1976, 40(1), 280–299 http://dx.doi.org/10.1016/0021-8693(76)90098-3[Crossref] | Zbl 0348.20017
[3] Huppert B., Endliche Gruppen I, Grundlehren Math. Wiss., 134, Springer, Berlin-New York, 1967 http://dx.doi.org/10.1007/978-3-642-64981-3[Crossref]
[4] Huppert B., Blackburn N., Finite Groups III, Grundlehren Math. Wiss., 243, Springer, Berlin-New York, 1982 http://dx.doi.org/10.1007/978-3-642-67997-1[Crossref]