Finite groups with an irreducible representation of large degree

In search of nonsolvable groups of central type

“…In the first case, inspection of the group order formulae leads to the possibilities: PSL (2, 8), PSL (3, 4), PSL(6, 2), PSP(6, 2), PΩ (5, 2), PΩ + (8, 2) and the solvable group PSU (3,2). For these groups, the primes r = 7, 5, 31, 5, 5 and 7 respectively satisfy hypothesis (1.1).…”

“…Iwahori and Matsumoto [10] conjectured that G must be solvable if A is central simple. DeMeyer and Janusz [2, Theorem 1] showed that such a group possesses a central extension G (of central type) such that there is a complex irreducible character X of G such that X(l) 2 …”

Characters fully ramified over a normal subgroup

“…In [12], G is shown to be solvable if G is of type f.r., and p B )f\G\ for any prime p dividing \G\. Suppose then H/Z is perfect, and let K/Z be a maximal normal subgroup.…”

“…Groups of type f.r. have been characterized in [12] as those groups G having a factor set a, over the multiplicative group of complex numbers, such that the corresponding twisted group algebra C[G] a is simple (or equivalently, has center = C). We shall have no occasion to use this characterization here.…”

“…1, Fully ramified triples* Groups of type Lr* To simplify notation, say that (H, N, χ) is a fully ramified triple if χ is an irreducible character of H, N is normal in H, and χ is fully ramified over N. It has been conjectured in [13] that H/N is solvable in this case, and some partial results in this direction appear in [12]. We extend this work in Theorem 4.5 below.…”