Vol. 58, No. 2, 1975

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On the group of permutations with countable support

Justin Thomas Lloyd and William Smiley

Vol. 58 (1975), No. 2, 529–530

Let SX denote the group of permutations of the set X. If α is an infinite cardinal, the set of permutations having support with cardinality less than or equal to α is a normal subgroup of SX. The principal result of this paper is a constructive proof that SX is generated by its cycles, if X is countably infinite. Of particular interest is the corollary that for any set X, the cycles of SX generate the subgroup of permutations with countable support.

Mathematical Subject Classification
Primary: 20F15
Received: 2 January 1974
Published: 1 June 1975
Justin Thomas Lloyd
William Smiley