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.
C. Lenart and N. Ray, “A Hopf algebraic framework for set system colourings with a group action,” preprint.
H. D.Macpherson, “A survey of Jordan groups”, Automorphisms of First-Order Structures (ed. R.Kaye and H. D.Macpherson), pp. 73–110, Oxford University Press, Oxford, 1994.
W.Rudin, “The automorphisms and the endomorphisms of the group algebra of the unit circle”, Acta Math. 95 (1956), 39–56.
H.Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.
About this article
Cite this article
Cameron, P.J. Cycle-closed permutation groups. J Algebr Comb 5, 315–322 (1996). https://doi.org/10.1007/BF00193181
- Permutation group
- Hopf algebra
- Fourier series