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We develop a general formula for the order of the group of automorphisms Aut(G) of a monolithic finite group G in terms of information about the complex characters of G and information about how G is embedded as a subgroup of a particular finite general linear group. We show that this formula is applicable to a wide variety of finite p-groups, and we discuss several applications to particular large...
If G is any finite solvable group having a normal Sylow 2-subgroup (in particular, if G is odd) and satisfying cd(G) >=5, where cd(G) is the set of ordinary irreducible character degrees of G, we show that the Fitting height of G does not exceed cd(G) -2. In case cd(G) =5, this upper bound on the Fitting height of G is best-possible.
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