A weak form of a conjecture on vanishing of group cohomology for blocks

TL;DR

This paper proves a weak version of a long-standing conjecture on the vanishing of group cohomology for blocks, using elementary module-theoretic lemmas.

Contribution

It introduces an elementary lemma related to pushout and pullback of modules and applies it to establish a partial result on a major open conjecture.

Findings

Proved a weak form of the conjecture on group cohomology vanishing.

Developed an elementary lemma involving pushout and pullback of modules.

Provided new insights into the structure of blocks in group cohomology.

Abstract

In this paper, we get an elementary and important lemma(See Lemma 3.2) which is about pushout and pullback of modules. And we prove a weak form of a long open conjecture on vanishing of group cohomology for blocks.