It is known that for finite simple groups it is possible for a faithful absolutely irreducible module to have 1-cohomology of dimension at least 3. However, until now no explicit examples have been found. We present two explicit examples where the dimension is exactly 3. It remains an open question as to whether the dimension can be bigger than 3.
In this article we show that the isomorphism type of certain semilinear classical groups may depend on the choice of form matrix, as well as the dimension and the field size. When there is more than one isomorphism type, we count them and present effective polynomial-time algorithms to determine whether two such groups are isomorphic.
In the Kourovka Notebook, Deaconescu asks if |Aut G| ≥ φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent.