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 automorphism of order p. This paper proves the conjecture for finite p-groups of coclass 4 and 5 (p >= 5). We also prove the conjecture for an odd order nonabelian p-group G with cyclic center satisfying CG(Gp gamma 3(G)) boolean AND Z3(G) <= Z(phi(G)).