Cycle-closed permutation groups
Peter J. Cameron
Queen Mary and Westfield College School of Mathematical Sciences Mile End Road E1 4NS London UK Mile End Road E1 4NS London UK
DOI: 10.1007/BF00193181
Abstract
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.
Pages: 315–322
Keywords: permutation group; cycle; Hopf algebra; Fourier series
Full Text: PDF
References
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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.