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.