On a criterion for solvability of a finite group

Abstract

Let G be a finite group and ${\sigma }_1(G)=\frac{1}{\left|G\right|}\sum _{H\le G}\left|H\right|.$ Answering to a problem posed by M. Tărnăuceanu, we prove that, if ${\sigma }_1(G)<\frac{117}{20},$ then G is solvable.

Keywords:

• Solvable groups
• subgroups orders

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 20D60
• Secondary 20D10
• 20F16

Additional information

Funding

This work was supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM), Italy.