On Sylow p-subgroups of finitary permutation groups

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2026年3月7日

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A. O. Asar

摘要

In this work it is shown that, for a prime number p, a totally imprimitive group of finitary permutations on an infinite set has either a unique Sylow p-subgroup or the cardinality of its Sylow p-subgroups is equal to $2^{\aleph _0}$ $2^{ℵ_{0}}$ . If in particular a Sylow p-subgroup of the group is transitive, then either the group is a p-group or the cardinality of its transitive Sylow p-subgroups is equal to $2^{\aleph _0}$ $2^{ℵ_{0}}$ . It follows from this that the cardinality of the Sylow p-subgroups of the finitary symmetric group on a infinite set is equal to $2^{\aleph _0}$ $2^{ℵ_{0}}$ .