jats:titleAbstract</jats:title> jats:pIn this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0134_ineq_0001.png" /> jats:tex-math\eta(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo 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-2022-0134_ineq_0002.png" /> jats:tex-math\eta(G/N)=\eta(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0134_ineq_0003.png" /> jats:tex-mathG/\langle G^{-}\rangle</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0134_ineq_0004.png" /> jats:tex-mathG^{-}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0134_ineq_0003.png" /> jats:tex-mathG/\langle G^{-}\rangle</jats:tex-math> </jats:alternatives> </jats:inline-formula> is either trivial, elementary abelian, a Frobenius group or isomorphic to jats:inline-formula jats:alternatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0134_ineq_0006.png" /> jats:tex-mathA_{5}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.</jats:p>