For large alternating groups, any three conjugacy classes of size at least |G|^(1-δ) multiply to the whole group for a universal small δ.
Squares of Conjugacy Classes in Alternating Groups
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abstract
We extend to alternating groups $A_n$ several results about symmetric groups asserting that under various conditions on a conjugacy class, or more generally, a normal subset, $C$ of $S_n$, we have $C^2 \supseteq A_n\setminus\{1\}$
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Products of three conjugacy classes in the alternating group
For large alternating groups, any three conjugacy classes of size at least |G|^(1-δ) multiply to the whole group for a universal small δ.