Geodesic
Cycle-closed permutation groups
Journal of Algebraic Combinatorics, Tome 5 (1996) no. 4, pp. 315-322.

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Summary: 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.
Keywords: permutation group, cycle, Hopf algebra, Fourier series 
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     author = {Cameron, Peter J.},
     title = {Cycle-closed permutation groups},
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Cameron, Peter J. Cycle-closed permutation groups. Journal of Algebraic Combinatorics, Tome 5 (1996) no. 4, pp. 315-322. http://geodesic.mathdoc.fr/item/JAC_1996__5_4_a4/