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

Document Type : Research Paper


Ms. Komma Patali Ph.D. Candidate School of Mathematics, IISER Thiruvananthapuram, Thiruvaanthapuram, India


A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner automorphism of order $p$. This paper proves the conjecture for finite $p$-groups of coclass $4$ and $5$ ($p\ge 5$). We also prove the conjecture for an odd order nonabelian $p$-group $G$ with cyclic center satisfying $C_G(G^p\gamma_3(G))\cap Z_3(G)\le Z(\Phi(G))$.


Main Subjects

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  • Receive Date: 22 November 2022
  • Revise Date: 06 April 2023
  • Accept Date: 08 April 2023
  • Published Online: 01 March 2024