Positions of characters in finite groups and the Taketa inequality

ABSTRACT

We define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.

KEYWORDS:

• Character degrees
• derived length
• solvable groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C15

Acknowledgments

I would like to thank Jørn B. Olsson for acting as advisor on my master’s thesis, in which the ideas of this paper first emerged. I would also like to thank Mark L. Lewis for reading an early version of the paper and providing helpful comments. Thanks to Alex Clark for helping in running some GAP code simultaneously on many computers. Finally, I would like to thank the anonymous referee for suggesting several helpful changes.