Solvability of Groups via Cyclic Subgroup Count

TL;DR

This paper introduces new criteria for determining the solvability and supersolvability of finite groups based on their count of cyclic subgroups, and extends the classification of n-cyclic groups for n≥13.

Contribution

It provides novel criteria linking cyclic subgroup counts to group solvability and advances the classification of n-cyclic groups for larger n.

Findings

Established new solvability criteria based on cyclic subgroup counts.

Extended classification of n-cyclic groups for n≥13.

Connected subgroup structure to group solvability properties.

Abstract

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially extends the classification of n-cyclic groups for n\geq 13.