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LetK  =   x  G{K=x^{G}}    be the conjugacy class of an element  x  of a group  G , and suppose  K  is finite. We study the increasing sequence of natural numbers{   |   K  n   |   }    n  ≥  1{\{\lvert K^{n}\rvert\}_{n\geq 1}}    and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if|   K  2   |   &lt;    3  2   ⁢   |  K  |{\lvert K^{2}\rvert&lt;\frac{3}{2}\lvert K\rvert}    or if|   K  4   |   &lt;   2  ⁢   |  K  |{\lvert K^{4}\rvert&lt;2\lvert K\rvert}   , thenK  n{K^{n}}    is a coset of the normal subgroup[  x  ,  G  ]{[x,G]}    for alln  ≥  2{n\geq 2}    or 4, respectively. We then use these results to contribute to conjectures about the solubility of〈  K  〉{\langle K\rangle}    whenK  n{K^{n}}    satisfies certain conditions.

Applying combinatorial results to products of conjugacy classes