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Cycle-closed permutation groups

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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.


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Cameron, P.J. Cycle-closed permutation groups. J Algebr Comb 5, 315–322 (1996).

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  • Permutation group
  • cycle
  • Hopf algebra
  • Fourier series