Some notes on defect groups (Q5929363)

First result: Let \(p\) be an odd, non-Mersenne prime, let \(G\) be a finite \(p\)-solvable group, and let \(D\) be a defect group of a \(p\)-block of \(G\). Then there are an element \(y\in O^{p'}(G)\cap C_G(D)\) and a Sylow \(p\)-subgroup \(P\) of \(G\) such that \(P\cap P^y=D\) and \(N_P(D)\) is a Sylow \(p\)-subgroup of \(N_G(D)\). The conditions on \(p\) are necessary, in general.NEWLINENEWLINENEWLINESecond result: Let \(G\) be a finite group, and let \(p\) be prime. If \(I_p\) is a Sylow \(p\)-subgroup of \(C_G(O^p(F^*(G)))\), then each defect group of each \(p\)-block of \(G\) contains a conjugate of \(I_p\).