Cyclic Probability of a Finite Group

The cyclic probability, $\theta (G)$, of a finite group $G$, is defined as the probability that two randomly selected elements of $G$ generate a cyclic subgroup of $G$. In this paper, we study various upper and lower bounds of $\theta (G)$ and show that $\theta (G)$ can be used as a criterion for checking nilpotency and solvability of a finite group. We also compare $\theta (G)$ with other group invariants like commuting probability $cp(G)$ and normalized sum of element orders.