jats:titleAbstract</jats:title> jats:pLet <jats:inline-formula id="j_jgth-2020-0036_ineq_9999_w2aab3b7e4213b1b6b1aab1c15b1b1Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>G</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0043.png" /> jats:tex-math{K=x^{G}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the conjugacy class of an element jats:italicx</jats:italic> of a group jats:italicG</jats:italic>, and suppose jats:italicK</jats:italic> is finite. We study the increasing sequence of natural numbers <jats:inline-formula id="j_jgth-2020-0036_ineq_9998_w2aab3b7e4213b1b6b1aab1c15b1b9Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:msup> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0168.png" /> jats:tex-math{{\lvert K^{n}\rvert}_{n\geq 1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if <jats:inline-formula id="j_jgth-2020-0036_ineq_9997_w2aab3b7e4213b1b6b1aab1c15b1c11Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:msup> <m:mi>K</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo><</m:mo> <m:mrow> <m:mfrac> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>K</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0118.png" /> jats:tex-math{\lvert K^{2}\rvert<\frac{3}{2}\lvert K\rvert}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or if <jats:inline-formula id="j_jgth-2020-0036_ineq_9996_w2aab3b7e4213b1b6b1aab1c15b1c13Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:msup> <m:mi>K</m:mi> <m:mn>4</m:mn> </m:msup> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo><</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>K</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0134.png" /> jats:tex-math{\lvert K^{4}\rvert<2\lvert K\rvert}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula id="j_jgth-2020-0036_ineq_9995_w2aab3b7e4213b1b6b1aab1c15b1c15Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0069.png" /> jats:tex-math{K^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a coset of the normal subgroup <jats:inline-formula id="j_jgth-2020-0036_ineq_9994_w2aab3b7e4213b1b6b1aab1c15b1c17Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0086.png" /> jats:tex-math{[x,G]}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula id="j_jgth-2020-0036_ineq_9993_w2aab3b7e4213b1b6b1aab1c15b1c19Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0184.png" /> jats:tex-math{n\geq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or 4, respectively. We then use these results to contribute to conjectures about the solubility of <jats:inline-formula id="j_jgth-2020-0036_ineq_9992_w2aab3b7e4213b1b6b1aab1c15b1c21Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0100.png" /> jats:tex-math{\langle K\rangle}</jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula id="j_jgth-2020-0036_ineq_9991_w2aab3b7e4213b1b6b1aab1c15b1c23Aa"> jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2020-0036_eq_0069.png" /> jats:tex-math{K^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies certain conditions.</jats:p>