Non-inner automorphisms of order p in finite p-groups of coclass 3

Abstract

In this paper we study the existence of at least one noninner automorphism of order p of a non-abelian finite p-group of coclass 3, whenever the prime p = 3.2010 Mathematics Subject Classification. 20D45, 20D15.

On the non-inner automorphisms of order $p$ in finite $p$-groups

“…In 2014, using derivation, Abdollahi et al [1] showed that every p-group of co-class 2 has a non-inner automorphism of order p leaving Z(G) elementwise fixed. Recently, Ruscitti et al [7] confirm the conjecture for finite p-group of co-class 3, with p = 3. Also, it is not very difficult to see that every finite p-group G of maximal class has a non-inner central automorphism of order p that fixes Φ(G) elementwise.…”

A note on non-inner automorphisms in finite normally constrained $p$-groups

“…To be more precise, Abdollahi and Ghoraishi in [4] proved that in some cases the non-inner automorphism of order p can be chosen so that it leaves Z(G) elementwise fixed. Finally, Abdollahi et al [5] proved the conjecture for p-groups of coclass 2, and quite recently in [17] M.Ruscitti, L. Legarreta and M.K.Yadav did the same for p-groups of coclass 3 when p is a prime different from 3.…”

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

“…Hence, (20) yields that G admits a non-inner automorphism of order p leaving G p γ 3 (G) elemnetwise fixed, whenever Z 5 (G)∩Z(G p γ 3 (G))…”

“…Abdollahi, Ghoraishi, and Wilkens [4] proved the conjecture for finite p-groups of class 3, and Abdollahi et al [5] proved the conjecture for p-groups of coclass 2. Ruscitti et al [20] proved the conjecture for p-groups of coclass 3 with the exception of p = 3. Ghoraishi [10,11] proved the conjecture for groups not satisfying the condition Z * 2 (G) ≤ C G (Z * 2 (G)) = Φ(G), and for an odd order p-group G for which (G, Z(G)) is a Camina pair.…”