A character-theoretic criterion for the existence of normal complements to subgroups of finite groups. (English) Zbl 0594.20007
A subgroup H of a finite group G is said to have the CR property in G if each irreducible complex character of H is the restriction of a character of G. H. Schmidt and F. Richen [J. Lond. Math. Soc., II. Ser. 7, 168-170 (1973; Zbl 0285.20025)] proved that if H has the CR property in a soluble group G and is a Carter subgroup of G then H has a normal complement in G. The authors generalize this to: Theorem. Let \({\mathcal F}\) be a saturated formation, let G be a finite soluble group, and let H be an \({\mathcal F}\)-projector of G. If H has the CR property in G, then H has a normal complement in G. This result is proved by first studying the properties of a minimal counterexample and then arriving to a contradiction.
Reviewer: B.G.Basmaji
MSC:
| 20C15 | Ordinary representations and characters |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D40 | Products of subgroups of abstract finite groups |
Keywords:
CR property; irreducible complex character; saturated formation; finite soluble group; normal complementCitations:
Zbl 0285.20025References:
| [1] | Carter, R. W.; Hawkes, T. O., TheJ-normalizers of a finite soluble group, J. Algebra, 5, 175-202 (1967) · Zbl 0167.29201 |
| [2] | Hawkes, T. O.; Jones, G. R., On the structure of a group whose orbits on a finite module are the orbits of a proper subgroup, J. Algebra, 94, 364-381 (1985) · Zbl 0601.20011 |
| [3] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0217.07201 |
| [4] | Isaacs, I. M., Character Theory of Finite Groups (1976), Academic Press: Academic Press New York/San Francisco/London · Zbl 0337.20005 |
| [5] | Richen, F. A.; Schmidt, H. J., A character-theoretic complementation theorem for Carter subgroups, J. London Math. Soc (2), 7, 168-170 (1973) · Zbl 0285.20025 |
| [6] | Sah, C. H., Existence of normal complements and extensions of characters in finite groups, Illinois J. Math., 6, 282-291 (1962) · Zbl 0105.25602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
