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Determining the Frattini Subgroup from the Character Table

Published online by Cambridge University Press:  20 November 2018

Sidney Garrison
Sidney Garrison*
Affiliation:
Texas A & M University, College Station, Texas
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Brauer [1, p. 141] has discussed the question of which subgroups of a group can be determined from its character table. He mentions, referring to p-groups, that the Frattini subgroup can be determined. We show that for an arbitrary finite solvable group, the Frattini subgroup can be determined from the character table. Then we exhibit an infinite set of pairs of non-solvable groups such that both members of a given pair have the same character table but Frattini subgroups of different orders. All groups to be considered are finite.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 560 - 567
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Brauer, R., Representations of finite groups, from Saaty, T. L., Lectures on modern mathematics, Vol. I (John Wiley & Sons, Inc., New York, 1963).Google Scholar
2. Feit, W., Characters of finite groups (W. A. Benjamin, Inc., New York, 1967).Google Scholar
3. Gallagher, P. X., Group characters and normal Hall subgroups, Nagoya Math. J. 21 (1962), 223–230.Google Scholar
4. Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, 1967).Google Scholar