Finite Groups Having Monolithic Characters of Prime Degree

Abstract

Let G be a finite group. An irreducible character χ is called monolithic when the factor group G/ker⁡(χ) has unique minimal normal subgroup. In this paper, we prove that for the smallest prime q dividing the order of G if G has a faithful imprimitive monolithic character of degree q, then G becomes a nonabelian q-group or a Frobenius group with cyclic Frobenius complement whose order is q. Under certain conditions, we also classify finite groups in which their nonlinear irreducible characters are monolithic.

Keywords

• Finite groups
• Monolithic characters
• Primitive characters

Asal Dereceli Monolitik Karakterlere Sahip Sonlu Gruplar

Abstract

G bir sonlu grup olsun. Bir χ indirgenemez karakterine, eğer G/ker⁡(χ) bölüm grubunun yalnız bir tane minimal normal alt grubu varsa monolitik karakter denir. Bu çalışmada, G grubunun mertebesini bölen en küçük asal sayı q olmak üzere, G derecesi q olan bir sadık imprimitif monolitik karaktere sahipse ya bir abelyen olmayan q-grubu ya da q mertebeli Frobenius tümleyeni olan bir Frobenius grubu olduğunu ispat ediyoruz. Lineer olmayan tüm indirgenemez karakterleri monolitik olan sonlu grupları da bazı koşullar altında sınıflandırıyoruz.

Keywords

• Finite Groups
• Monolithic Characters
• Primitive Characters

References

1. [1] Y.G. Berkovich and E.M. Zhmud, Characters of Finite Groups Part 2, American Mathemetical Society, 1999.
2. [2] Y.G. Berkovich, “On Isaacs' three character degrees theorem”, Proc. Am. Math. Soc. vol. 125, no. 3, pp. 669-677, 1997 .
3. [3] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967.
4. [4] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976.
5. [5] I.M. Isaacs, Algebra: A Graduate Course, Brooks/Cole, Pacific Grove, CA, 1994.
6. [6] O. Manz and T.R. Wolf, Representations of Solvable Groups, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993.