Suppose that we have a finite group $G$ and choose elements $a, b \in G$ at random. What is knowncan be said about the mean order or index of athe subgroup of some finite group generated by two elements chosen at random$a$ and $b$? Mainly, what is the expected order, $\epsilon_1(G)$, and index of such a subgroup?
EDIT: I understand that the question is too general in the present setting. Therefore, splitting the question up might be a good idea.
For which groups $G$ is the mean order easily computable? The comments suggest that $\mathbb Z_n$ and $S_n$ belong to that category.
Are there any asymptotic results available, depending on $|G|$? While $\epsilon_1(G)$ or $\sum_{|G|=n} \epsilon_1(G)$ might fluctuate wildly, $\sum_{|G|\leq n} \epsilon_1(G)$ could be more tractable.
The mean order is the arithmetic mean of the orders of all 2-generator (+ the trivial and 1-generator) subgroups. The mean index is the harmonic mean of orders of all 2-generator subgroups, so that we may suggestively denote it by $\epsilon_{-1}(G)$. And similarly, we can look at other expected values, $\epsilon_\alpha(G)=\sum_{a,b\in G} |\langle a,b \rangle|^\alpha$. Are there any results, in the above senses, available for these values?
