On some problems about automorphisms of finite p-groups

Three important problems in the context of automorphisms of finite p-groups are as follows. First, does every finite nonabelain p-group admit a noninner automorphism of order p? Second, does the order of every finite p-group G divide the order of its automorphisms group, Aut(G)? Third, does there exist any finite p-group G, other that D8, such that G ∼ = Aut(G)? This talk includes a brief survey of the recent developments concerning these problems. For the last few years, the speaker was interested in these problems and had a modest contribution in the first one