A variant of a theorem of J. A. Green
A classical theorem of \textit{J. A. Green} [Math. Z. 70, 430-445 (1959; Zbl 0086.024)] asserts that for a given finite group G with a normal subgroup N such that G/N is a p-group, and an absolutely indecomposable KN-module V, where K is a field of characteristic p, the induced module \(V^ G\) is absolutely indecomposable. The present article studies a similar situation without assuming V to be absolutely indecomposable. Theorem: Let K be a perfect field with characteristic \(p>0\), let \(N\trianglelefteq G\) and let V be an indecomposable KN-module whose inertia group modulo N is a p-group. Then all indecomposable components of \(V^ G\) are isomorphic. An elementary example \((K=GF(2)\), \(G=\Sigma_ 3\), \(N=A_ 3)\) is given showing that \(V^ G\) need not be indecomposable.
scientific article; zbMATH DE number 3979339