Vol. 39, No. 2, 1971

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Permutations as products of conjugate infinite cycles

Edward Arthur Bertram

Vol. 39 (1971), No. 2, 275–284

Let S = {ai}−∞ be a countable set and P any permutation of S with infinite support. Since the subgroup generated by the conjugacy class 𝒦 of P must be normal in Sym(S), we know that every permutation of S is a product of permutations from 𝒦 Since it has recently been discovered that every even permutation in the finite symmetric group Sym(n) may be expressed as a product of exactly two n-cycles, we are naturally led to a similar question for Sym (S) and the infinite cycle C = (,a2,a1,a0,a1,a2,), with support all of S. In this paper it is proved that for each k 3 every permutation of S is a product of exactly k cycles conjugate to C, but that no odd finite permutation is a product of two.

Mathematical Subject Classification 2000
Primary: 20B10
Received: 10 November 1970
Published: 1 November 1971
Edward Arthur Bertram