non-inner automorphisms of order p in finite p-groups of coclass 4 and 5

a long-standing conjecture asserts that every finite nonabelian p-group has a non-inner automorphismof order p. this paper proves the conjecture for finite p-groups of coclass 4 and 5 (p ≥ 5). we also provethe conjecture for an odd order nonabelian p-group g with cyclic center satisfying cg(gpγ3(g)) ∩ z3(g) ≤z(φ(g)).