Conjugacy classes whose square is an infinite symmetric group

Author:
Gadi Moran

Journal:
Trans. Amer. Math. Soc. **316** (1989), 493-522

MSC:
Primary 20B07

DOI:
https://doi.org/10.1090/S0002-9947-1989-1020501-5

MathSciNet review:
1020501

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Abstract: Let be the set of all permutations of an infinite set of cardinality with the property: every permutation of is a product of two conjugates of . The set is shown to be the set of permutations satisfying one of the following three conditions:

(1) has at least two infinite orbits.

(2) has at least one infinite orbit and infinitely many orbits of a fixed finite size .

(3) has: no infinite orbit; infinitely many finite orbits of size and for some positive integers ; and infinitely many orbits of size . It follows that iff some transposition is a product of two conjugates of , and is not a product , where has a finite support and is an involution.

For iff moves elements, and satisfies (1), (2) or , where is obtained from (3) by omitting the requirement that has infinitely many orbits of size . It follows that for iff moves elements and some transposition is the product of two conjugates of .

The covering number of a subset of a group is the smallest power of (if any) that equals [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.

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DOI:
https://doi.org/10.1090/S0002-9947-1989-1020501-5

Article copyright:
© Copyright 1989
American Mathematical Society