On the cohomological triviality of the center of the Frattini subgroup

TL;DR

This paper investigates the properties of potential counterexamples to a conjecture related to the Frattini subgroup's center, providing new bounds, classifications, and confirming the conjecture for specific classes of groups.

Contribution

It improves lower bounds on counterexample orders, characterizes minimal counterexamples, and verifies the conjecture for certain classes of p-groups.

Findings

Lower bounds on counterexample orders are improved.

Properties of minimal counterexamples are determined.

The conjecture holds for specific classes of nonabelian p-groups.

Abstract

We improve the existing lower bounds on the order of counterexamples to a conjecture by P. Schmid, determine some properties of the possible counterexamples of minimum order for each prime, and the isomorphism type of the center of the Frattini subgroup for the counterexamples of order 256. We also show that nonabelian metacyclic p-groups, nonabelian groups of maximal nilpotency class and 2-groups of coclass two satisfy the conjecture.