Symmetric groups and conjugacy classes
Edith Adan-Bante, Helena Verrill

TL;DR
This paper investigates the structure of conjugacy class products in symmetric groups, proving they are never a single conjugacy class and characterizing when they form unions of two or more classes.
Contribution
It provides new results on the behavior of conjugacy class products in symmetric groups, including conditions for when these products form unions of multiple classes.
Findings
Product of two nontrivial conjugacy classes is never a single class.
If n is not divisible by 2 or 3, the product is at least three classes.
Characterization of elements when the product is exactly two classes.
Abstract
Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial , we prove that the product of the conjugacy classes and is never a conjugacy class. Furthermore, if n is not even and is not a multiple of three, then is the union of at least three distinct conjugacy classes. We also describe the elements in the case when is the union of exactly two distinct conjugacy classes.
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Taxonomy
TopicsFinite Group Theory Research · Advanced Algebra and Geometry · Geometric and Algebraic Topology
