Blocks of small defect. (Q2341307)

One of the main results of the paper is the following. Let \(G\) be a finite solvable group. Let \(p\) be a prime such that \(p\geq 5\) and \(O_p(G)=1\), and write \(|G|_p=p^n\). Then \(G\) contains a block of defect less than or equal to \(\lfloor\frac{3n}{5}\rfloor\). This constitutes progress towards a conjecture of Espuelas and Navarro who had suggested that the result should hold for arbitrary groups with \(1/2\) in place of \(3/5\). The proof of this result is based on the following strong result on orbits which is also proved in the paper. Theorem. Let \(\pi_0\) be the set of all the primes except 2 and 3. Let \(G\) be a finite solvable group and let \(V\) be a finite, faithful and completely reducible \(G\)-module (possibly of mixed characteristic). Then there exists a normal subgroup \(K\) of \(G\) with \(K\leq\mathbf F_2(G)\) and there exist two \(G\)-orbits with representatives \(v_a,v_b\in V\) such that for any Hall \(\pi_0\)-group \(H\) we have \(C_H(v_a)\leq K\) and \(C_H(v_b)\leq K\). Moreover, the Hall \(\pi_0\)-subgroup of \(K\mathbf F(G)/\mathbf F(G)\) and the Hall \(\pi_0\)-subgroup of \(K\cap\mathbf F(G)\) are abelian. This theorem is also used to prove that if \(G\) is a finite solvable group and \(p^a\) is the largest power of \(p\) dividing \(\chi(1)\) for an irreducible character \(\chi\) of \(G\), then \(|G:\mathbf F(G)|_p\leq p^{3a}\) for \(p\geq 5\).