Let G be a finite non-abelian p-group, where p is a prime. Let Autc(G) and Autz(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Autc(G) = Autz(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Autc(G) = Autz(G). We also characterize all finite p-groups G of order ≤ p 7 such that Autc(G) = Autz(G) and complete the classification of all finite pgroups of order ≤ p 5 for which there exist non-inner class preserving automorphisms.

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An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G′. Let IA (G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA (G) ≃ Inn (G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner.

Abstract. We study finite p-groups G of coclass upto 4 for which the group Autz(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].