A Character Theoretic Condition for F(G) > 1

ABSTRACT

If G is a nontrivial finite group with the property that χ (1) ² divides bi| G | for all irreducible (complex) characters χ , then G has a nontrivial normal abelian subgroup.

^#Communicated by A. Turull.

Key Words:

• Fitting subgroup
• Irreducible characters

Mathematics Subject Classification:

• Primary 20C15
• Secondary 20D25

ACKNOWLEDGMENT

I would like to acknowledge the help of my wife Gloria, in whose memory this paper is dedicated. She found all of the partitions λ in the first table appearing in the proof of Lemma (3.2), and Table 2, allowing me to concentrate on the argument in the large. Her greatest measure came later, providing inspiration by her example of unwavering courage and faith in the face of increasingly overwhelming odds.

Notes

^#Communicated by A. Turull.

p ‐ parts of hook numbers of selected partitions (p off).

2 ‐ parts of hook numbers of selected partitions (small cases).

Character with large 2 ‐ part degrees in selected sporadic groups.