The rank of the automorphism group of a finite group (Q2281524)

Let \(G\) be a finite group whose order is divisible by \(e\) primes (counting repetitions). Then the automorphism group of \(G\) has Prüfer rank at most \(e+[e^2/2],\) meaning that each subgroup of the automorphism group of \(G\) can be generated by at most \(e+[e^2/2],\) elements. The proof uses the following result, of independent interest. Let \(G\) be a finite subgroup of \(\mathrm{GL}(n, F),\) where \(n\) is a positive integer and \(F\) is a field of positive characteristic. If \(G\) has a soluble normal subgroup \(S\) such that \(G/S\) is a direct product of \(r\) perfect simple groups, then \(r \leq [n/2].\)