2021 Lower central words in finite $p$-groups
Iker de las Heras, Marta Morigi
Publ. Mat. 65(1): 243-269 (2021). DOI: 10.5565/PUBLMAT6512107


It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $\gamma_r$ and for $p\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $\gamma_r(G)$ is abelian and $2$-generator, then $\gamma_r(G)$ consists only of $\gamma_r$-values. In this paper we extend this result, showing that the assumption that $\gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p\!=\!3$. Finally, we prove that the analogous result for pro-$p$ groups is true.

Funding Statement

The first author is supported by the Spanish Government, grant MTM2017-86802-P, partly with FEDER funds, and by the Basque Government, grant IT974-16. He is also supported by a predoctoral grant of the University of the Basque Country. The second author is a member of INDAM.


Download Citation

Iker de las Heras. Marta Morigi. "Lower central words in finite $p$-groups." Publ. Mat. 65 (1) 243 - 269, 2021. https://doi.org/10.5565/PUBLMAT6512107


Received: 19 September 2019; Revised: 10 March 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185832
Digital Object Identifier: 10.5565/PUBLMAT6512107

Primary: 20D15
Secondary: 20F12 , 20F14

Keywords: $p$-groups , commutators , lower central words

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques


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Vol.65 • No. 1 • 2021
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