On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

Yiftach Barnea; Rachel Camina; Mikhail Ershov; Mark L. Lewis

On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

Group Theory 2025-02-07 v3 Number Theory

Authors: Yiftach Barnea , Rachel Camina , Mikhail Ershov , Mark L. Lewis

Abstract

Given a discrete (resp. profinite) group $G$ , we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$ . In this paper we determine all residually finite discrete groups with finite NCC and give an almost complete characterization of profinite groups with finite NCC.

Keywords

covering by conjugates cyclic subgroups procyclic subgroups residually finite groups profinite groups NCC

Cite

@article{arxiv.2210.15746,
  title  = {On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups},
  author = {Yiftach Barnea and Rachel Camina and Mikhail Ershov and Mark L. Lewis},
  journal= {arXiv preprint arXiv:2210.15746},
  year   = {2025}
}

Comments

v3: 30 pages, to appear in Math. Annalen. Major revision based on the referee's report. Sections 6 and 7 have been completely rewritten. We have also removed the results from the original paper dealing with NCC for families of finite p-groups or relied on computations in the group PGL_1(D) where D is the quaternion division algebra over Q_p. We plan to include those results in a follow-up paper

On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

Abstract

Keywords

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