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Inverse semigroups with idempotent-fixing automorphisms

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Abstract

A celebrated result of J. Thompson says that if a finite group \(G\) has a fixed-point-free automorphism of prime order, then \(G\) is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

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Acknowledgments

We would like to thank the anonymous referee for suggesting the proof of the second part of Lemma 2 in response to a conjecture in an earlier draft of this paper. We are pleased to acknowledge the assistance of the automated theorem prover Prover9 and the finite model builder Mace4, both developed by W. McCune [4]. The first author was partially supported by FCT through PEst-OE/MAT/UI0143/2011.

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Authors and Affiliations

  1. Universidade Aberta, 1269-001 , Lisbon, Portugal

    João Araújo

  2. Centro de Álgebra, Universidade de Lisboa, 1649-003 , Lisbon, Portugal

    João Araújo

  3. Department of Mathematics, University of Denver, 2280 S Vine St, Denver, CO , 80208, USA

    Michael Kinyon

Authors
  1. João Araújo
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  2. Michael Kinyon
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Correspondence to João Araújo.

Additional information

Communicated by Mark V. Lawson.

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Araújo, J., Kinyon, M. Inverse semigroups with idempotent-fixing automorphisms. Semigroup Forum 89, 469–474 (2014). https://doi.org/10.1007/s00233-014-9585-0

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  • DOI: https://doi.org/10.1007/s00233-014-9585-0

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