Word automatic groups of nilpotency class 2

We consider word automaticity for groups that are nilpotent of class 2 and have exponent a prime p. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special p-group Ep is word automatic, as well as an intermediate group Hp which...

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Bibliographic Details
Published in:Information processing letters 2024-01, Vol.183, p.106426, Article 106426
Main Authors: Nies, André, Stephan, Frank
Format: Article
Language:English
Subjects:
formal language
nilpotent group
word automaticity
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Summary:We consider word automaticity for groups that are nilpotent of class 2 and have exponent a prime p. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special p-group Ep is word automatic, as well as an intermediate group Hp which has an infinite centre. In the last section we introduce a method for showing automaticity of central extensions of abelian groups via co-cycles. •We study word automaticity in the variety of groups of nilpotency class 2 and exponent p.•The free group of infinite rank is not word automatic.•An infinite extra-special p-group is word automatic.•We introduce a construction of word-automatic groups via co-cycles.
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2023.106426