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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ {X_\nu }$ be the set of all permutations $ \xi $ of an infinite set $ A$ of cardinality $ {\aleph _\nu }$ with the property: every permutation of $ A$ is a product of two conjugates of $ \xi $. The set $ {X_0}$ is shown to be the set of permutations $ \xi $ satisfying one of the following three conditions:

(1) $ \xi $ has at least two infinite orbits.

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

(3) $ \xi $ has: no infinite orbit; infinitely many finite orbits of size $ k,l$ and $ k + l$ for some positive integers $ k,l$; and infinitely many orbits of size $ > 2$. It follows that $ \xi \in {X_0}$ iff some transposition is a product of two conjugates of $ \xi $, and $ \xi $ is not a product $ \sigma i$, where $ \sigma $ has a finite support and $ i$ is an involution.

For $ \nu > 0,\;\xi \in {X_\nu }$ iff $ \xi $ moves $ {\aleph _\nu }$ elements, and satisfies (1), (2) or $ (3')$, where $ (3')$ is obtained from (3) by omitting the requirement that $ \xi $ has infinitely many orbits of size $ > 2$. It follows that for $ \nu > 0,\;\xi \in {X_\nu }\;$ iff $ \xi $ moves $ {\aleph _\nu }$ elements and some transposition is the product of two conjugates of $ \xi $.

The covering number of a subset $ X$ of a group $ G$ is the smallest power of $ X$ (if any) that equals $ G$ [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