Examples of non-FSZ p-groups for primes greater than three

TL;DR

This paper constructs specific non-FSZ p-groups for primes greater than three, demonstrating minimal order examples and expanding understanding of p-group structures.

Contribution

It provides explicit constructions of non-FSZ p-groups of minimal order for primes greater than three, filling a gap in group theory literature.

Findings

Constructed non-FSZ p-groups of order p^{p+1} for primes p>3.

Established minimal order for non-FSZ p-groups.

Extended the classification of p-groups beyond previously known cases.

Abstract

For any prime $p>3$ and $j\in\mathbb{N}$ we construct examples of non-$FSZ_{p^j}$ groups of order $p^{p^j+2j-1}$. In the special case of $j=1$ this yields groups of order $p^{p+1}$, which is the minimum possible order for a non-$FSZ$ $p$-group.