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.
Lower central words in finite p-groups / de las Heras I.; Morigi M.. - In: PUBLICACIONS MATEMÀTIQUES. - ISSN 0214-1493. - STAMPA. - 65:1(2021), pp. 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.| File | Dimensione | Formato | |
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