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 |
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