Cycle-closed permutation groups

Peter J. Cameron*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


A finite permutation group is cycle-closed if it contains all the cycles of all of its elements. It is shown by elementary means that the cycle-closed groups are precisely the direct products of symmetric groups and cyclic groups of prime order. Moreover, from any group, a cycle-closed group is reached in at most three steps, a step consisting of adding all cycles of all group elements. For infinite groups, there are several possible generalisations. Some analogues of the finite result are proved.

Original languageEnglish
Pages (from-to)315-322
Number of pages8
JournalJournal of Algebraic Combinatorics
Issue number4
Publication statusPublished - 1 Dec 1996


  • Cycle
  • Fourier series
  • Hopf algebra
  • Permutation group


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