A remark on socles and normal subgroups
Let k be an algebraically closed field and K a normal subgroup of the finite group G. Let W be an indecomposable kK-module, which is stable in G. It is shown that if W has a simple socle then so has any indecomposable direct summand of \(W^ G\) and the number of indecomposable summands of \(W^ G\) equals that of \((socle(W))^ G\). If W has a simple socle and if the characteristic of k is zero or prime to \(| G/K|\), then W is extendable to G iff socle(W) is. By passing to the contragredient modules one has the corresponding results for heads.
scientific article; zbMATH DE number 3857351