Small representations are completely reducible
The author's main result is the following. Let \(p\) be a prime and let \(k\) be a field of characteristic \(p\). Let \(G\) be a finite group that contains no non-trivial normal \(p\)-subgroup. Let \(V\) be a faithful \(kG\)-module. Then \(V\) is completely reducible if \(\dim V\leq p-2\). It follows then that first cohomology group \(H^1(G,V)\) is trivial if \(\dim V\leq p-3\). The author also notes that the alternating group \(A_p\) has an irreducible module of dimension \(p-2\) over \(k\) whose first cohomology group has dimension \(1\) as a vector space over \(k\), so the result above is best possible. The proof of this theorem is based on the following result (Theorem B of the paper). Let \(V\) be a faithful \(kG\)-module of dimension at most \(p-3\), where \(G\) is a finite group with no non-trivial normal \(p\)-subgroup. Assume that \(G\) is generated by its elements of order \(p\). Then either \(G\) is a central product of finite quasisimple Chevalley groups of characteristic \(p\) or \(p=11\), \(n=7\) or 8, and \(G\) is the smallest Janko group \(J_1\) (a case which can occur). The classification of finite simple groups is required for this result, together with results on the minimal dimensions of non-trivial modules of finite groups of Lie type. (The enunciation of Theorem B in the paper is incorrect, as it demands that the dimension is at most \(p+3\), rather than \(p-3\).) The author also includes versions of his theorems that hold for algebraic groups, with normal unipotent subgroups replacing normal \(p\)-subgroups.
scientific article; zbMATH DE number 1384474
scientific article