On equality of central and class preserving automorphisms of finite p-groups

Abstract

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.

Class-preserving automorphisms of some finite p-groups

“…Since G is of maximal class, |Z(Inn(G))| = p, |G | = p 3 , d(G) = 2 and hence |Aut z (G)| = |Hom(C p × C p , C p )| = p 2 . Also, since As a consequence of Theorem 2.3, we can now obtain the following results of Yadav ( §5 of [12]) and Kalra and Gumber (Theorem 4.2 of [7]). In [6], the groups of order p 5 , where p is an odd prime, are divided into ten isoclinism families; and in [4], the groups of order 32 are divided into eight isoclinism families.…”

“…In §2, we prove our main theorem, Theorem 2.3, which gives the necessary and sufficient conditions on a finite p-group G of order p 5 for which Out c (G) = 1. As a consequence, we obtain short and alternate proofs of the results of Yadav ( §5 of [12]) and Kalra and Gumber (Theorem 4.2 of [7]). …”

“…He then showed, by picking up one group from each isoclinism family, that Out c (G) = 1 for the groups 7 (1 5 ) and 10 (1 5 ) from seventh and tenth family in [6], and hence concluded that if G is a finite p-group of order p 5 , where p is an odd prime, then Out c (G) = 1 if and only if G is isoclinic to a group either in the seventh or in the tenth family. Recently, Kalra and Gumber (Theorem 4.2 of [7]), using the classifications, isoclinism families and presentations given by Hall and Senior [4] and Sag and Wamsley [10], have shown that if G is a finite 2-group of order 2 5 , then Out c (G) = 1 except for the forty fourth and forty fifth groups from the sixth family in [4].…”

Class-preserving automorphisms of some finite $p$-groups

“…In section 2, we prove our main theorem, Theorem 2.3, which gives the necessary and sufficient conditions on a finite p-group G of order p 5 for which Out c (G) = 1. As a consequence, we obtain short and alternate proofs of the results of Yadav [12, Section 5] and Kalra and Gumber [7,Theorem 4.2].…”

On Equality of Certain Automorphism Groups

“…Recently, many mathematicians got interested to study the equality of certain subgroups of an automorphism group and a number of results have been proved in this direction. Many interesting results have been proved in [5], [8], [6], [9] etc.. Most of the results, which have been proved in this direction, are either for a finite group or for a finite non-abelian p-group of nilpotency class at most two.…”