Conjugacy classes of small size in solvable groups

Abstract

If G is a finite group and $x\in G$, we say that x lies in a small class if $\left|x^G\right|$ is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra.

Keywords:

• Conjugacy class size
• eigenspace
• fitting subgroup
• Jordan form
• solvable group

2020 Mathematics Subject Classification:

• Primary: 15A18
• 15A20
• 20D10
• 20E45

Additional information

Funding

I thank R. Guralnick for Example 2.6 and J. Medina for many helpful conversations on this problem. Research supported by Ministerio de Ciencia e Innovación (Grant PID2019-103854GB-I00 funded by MCIN/AEI/10.13039/501100011033) and CIAICO/2021/163.