Cohomology, fusion and a p-nilpotency criterion

TL;DR

This paper establishes criteria for p-nilpotency of finite groups based on fusion control of cyclic subgroups, with specific conditions for odd and even primes, linking cohomology and fusion properties.

Contribution

It provides a new fusion control criterion for p-nilpotency involving cyclic groups of order p and 4, extending previous results.

Findings

G is p-nilpotent if and only if P controls fusion of cyclic groups of order p for odd p.

G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4 for p=2.

The criteria connect cohomological properties with fusion control in finite groups.

Abstract

Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p=2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.