# Finite non-cyclic $p$-groups whose number of subgroups is minimal

## TL;DR

This paper characterizes non-cyclic finite p-groups with the minimal number of subgroups among all p-groups of the same order, complementing previous work on maximal subgroup counts.

## Contribution

It provides a complete classification of non-cyclic finite p-groups with the minimal number of subgroups, filling a gap in subgroup enumeration.

## Key findings

- Identifies all non-cyclic p-groups with minimal subgroup counts
- Complements existing results on maximal subgroup counts
- Provides explicit structural descriptions

## Abstract

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by determining completely the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.