Noninner automorphisms of order \(p\) of finite \(p\)-groups (Q1602022)

The central result of the paper is the following theorem: Let \(G\) a finite nonabelian \(p\)-group such that \(\Phi(G)\neq C_G(Z(\Phi(G)))\). Then \(G\) has a noninner automorphism which fixes \(\Phi(G)\). This result reduces the verification of the conjecture that every finite nonabelian \(p\)-group has a noninner automorphism of order \(p\) to the case when \(\Phi(G)=C_G(Z(\Phi(G)))\). This conjecture is the sharpened version of the well-known theorem of Gaschütz asserting that any finite nonabelien \(p\)-group has a noninner automorphism of \(p\)-power order.