Linear maps over abelian group algebras (Q1903693)

Assume that \(K\) is a field of prime characteristic \(p\) and let \(G\) be a finite abelian \(p\)-group. The authors examine the following problem. Problem. Given a free \(KG\)-module \(V\) and \(\alpha\in\text{End}_{KG}(V)\), does there exist an integer \(d>0\) (depending on \(G\)) such that \(\dim_K\text{Ker }\alpha \geq d\) if \(\alpha\) is not an isomorphism? The result of the paper provides an affirmative answer. Namely, the following holds. Theorem. If \(V\) is finitely generated free \(KG\)-module and \(\alpha:V\to V\) is a noninjective homomorphism of \(KG\)-modules, then \(\dim_K\text{Ker }\alpha\geq|G|/(\text{exp}(G))\). A number of consequences is also provided.