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Noninner automorphisms of order p for finite p-groups of restricted coclass

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Abstract

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper, we give a lower bound for the coclass of finite nonabelian p-groups G having no noninner automorphism of order p leaving the Frattini subgoup \(\Phi (G)\) elementwise fixed. As a consequence, the verification of the conjecture is reduced to the case of finite nonabelian p-groups G in which the coclass of G is greater than the minimum number of generators of G.

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Acknowledgements

The authors would like to thank the referees for their valuable comments. This research work was done when the author was on sabbatical leave at University of Isfahan. The author would like to thank the Research Council of Shahid Chamran University of Ahvaz for their financial support (Grant Number: SCU.MM99.666).

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Authors and Affiliations

  1. Department of Mathematics, Shahid Chamran University of Ahvaz, 6135783151, Ahvaz, Iran

    S. Mohsen Ghoraishi

  2. Department of Pure Mathematics Faculty of Mathematics and Statistics, University of Isfahan, 81746-73441, Isfahan, Iran

    S. Mohsen Ghoraishi

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  1. S. Mohsen Ghoraishi
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Correspondence to S. Mohsen Ghoraishi.

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Ghoraishi, S.M. Noninner automorphisms of order p for finite p-groups of restricted coclass. Arch. Math. 117, 361–368 (2021). https://doi.org/10.1007/s00013-021-01647-4

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  • DOI: https://doi.org/10.1007/s00013-021-01647-4

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