2011/10/31 by Abdollahi, Alireza, Ghoraishi, Mohsen
#20D15 #20D45 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1110.6888
A longstanding conjecture asserts that every non-abelian finite p-group G admits a non-inner automorphism of order p. The conjecture is valid for finite p-groups of class 2. Here, we prove every finite non-abelian p-group G of class 3 with p>2 has a noninner automorphism of order p leaving Φ(G) elementwise fixed. We also prove that if G is a finite 2-group of class 3 which cannot be generated by 4 elements, then G has a non-inner automorphism of order 2 leaving Φ(G) elementwise fixed. We also prove that the latter conclusion holds for finite 2-groups G of class 3 such that the center of G is not cyclic and the minimal number of generators of G is 2 or 4 and it holds whenever the center of G is \em not 2-generated and the minimal number of generators of G is 3. Some results are also proved for the existence of non-inner automorphisms of order p for a finite p-group G under conditions in terms of the minimal number of generators of the center factor of G and a certain function of the rank of G.