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Finite 𝑝-groups of class two with a small multiple holomorph

  • A. Caranti ORCID logo and Cindy (Sin Yi) Tsang ORCID logo EMAIL logo
Published/Copyright: September 6, 2023

Abstract

We consider the quotient group T⁒(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T⁒(G) contains a cyclic subgroup of order prβˆ’1⁒(pβˆ’1), where pr is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with r=1) such that T⁒(G) has order exactly pβˆ’1, which is as small as possible.

Award Identifier / Grant number: 21K20319

Funding statement: The first-named author is a member of GNSAGA-INdAM, Italy. Both authors acknowledge that this research was supported by JSPS KAKENHI Grant Number 21K20319.

Acknowledgements

The first-named author acknowledges support from the Department of Mathematics of the University of Trento. The authors also thank the referee for helpful comments.

  1. Communicated by: Andrea Lucchini

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Received: 2023-04-11
Revised: 2023-07-10
Published Online: 2023-09-06
Published in Print: 2024-03-01

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