Nonexistence of perfect locally finite minimal non-FC and CC-groups, and a characterization of perfect locally finite barely transitive groups

ARIKAN A.

Ricerche di Matematica, 2026 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2026
  • Doi Number: 10.1007/s11587-025-01048-7
  • Journal Name: Ricerche di Matematica
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: Barely transitive group, CC-group, FC-group, Locally finite group, Minimal non-CC-group, Minimal non-FC-group, Perfect group
  • Gazi University Affiliated: Yes

Abstract

In this short note we prove that if G is a perfect locally finite barely transitive group, then the finitary residual of G, namely F(G)={g∈G||supp(g)|<ω}, is trivial. In particular, we prove that there do not exist any perfect locally finite minimal non-FC and minimal non-CC-groups. This completes the description of minimal non-FC-groups and locally graded minimal non-CC-groups. It is a long-standing problem (see [14, Problem 5.1(b)]).