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

Leire Legarreta Solaguren (University of the Basque Country)

Frank Adams 1,

The motivation for this talk is to contribute to a longstanding conjecture of Berkovich posed in 1973. It states that every finite \(p\)-group admits a non-inner automorphism of order \(p\), where \(p\) denotes a prime number. This conjecture can be viewed as a refinement of the celebrated theorem of Gascuetz that states that every non-abelian finite \(p\)-group admits a non-inner automorphism of order some power of \(p\). The conjecture has attracted the attention of many mathematicians during the last couple of decades, and has been confirmed for many interesting classes of finite \(p\)-groups. It is interesting to note that, in 1965, Liebeck proved the existence of a non-inner automorphism of order \(p\) in all finite \(p\)-groups of class 2, where \(p\) is an odd prime. For \(p=2\), he proved the existence of a non-inner automorphism of order 2 or 4. The fact that there always exists a non-inner automorphism of order 2 in all finite 2-groups of class 2 was proved by Abdollahi in 2007 and the conjecture was confirmed for finite regular \(p\)-groups by Schmid in 1980. Deaconescu proved it for all finite \(p\)-groups \(G\) which are not strongly Frattinian, i.e. \(C_G(Z(\Phi(G)))=\Phi(G)\). Abdollahi proved it for finite \(p\)-groups \(G\) such that \(G/Z(G)\) is a powerful \(p\)-group, and Jamali and Visesh proved the conjecture for finite \(p\)-groups with cyclic commutator subgroup. Quite recently, the conjecture has been confirmed for \(p\)-groups of nilpotency class 3 by Abdollahi, Ghoraishi and Wilkens, and for \(p\)-groups of coclass 2 by Abdollahi et al. For semi-abelian \(p\)-groups, the conjecture has been confirmed by Benmoussa and Guerboussa. In the talk I will add an important class of \(p\)-groups to the above list by proving that the above mentioned conjecture holds true for all finite \(p\)-groups of coclass 3 when \(p\neq 3\).

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