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 γr and for p ≥ 5, Guralnick showed that if G is a finite p-group such that the verbal subgroup γr(G) is abelian and 2-generator, then γr(G) consists only of γr-values. In this paper we extend this result, showing that the assumption that γr(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.

de las Heras I., Morigi M. (2021). Lower central words in finite p-groups. PUBLICACIONS MATEMÀTIQUES, 65(1), 243-269 [10.5565/PUBLMAT6512107].

Lower central words in finite p-groups

Morigi M.
2021

Abstract

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 γr and for p ≥ 5, Guralnick showed that if G is a finite p-group such that the verbal subgroup γr(G) is abelian and 2-generator, then γr(G) consists only of γr-values. In this paper we extend this result, showing that the assumption that γr(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.
2021
de las Heras I., Morigi M. (2021). Lower central words in finite p-groups. PUBLICACIONS MATEMÀTIQUES, 65(1), 243-269 [10.5565/PUBLMAT6512107].
de las Heras I.; Morigi M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/789435
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