Let $G$ be a Group. The set of element orders of $G$ and the set of numbers of the same order elements in $G$ are denoted by $π_e(G)$ and $τ_e(G)$ (sometimes nse($G$)), respectively. Let $π(n)$ be the set of prime divisors of $n$ and $π(G) = π(|G|)$ if $G$ is finite. In 1987, J.G.Thompson posed a very interesting problem related to algebraic number fields as follows.
Problem: Let $T(G)=\lbrace (n,s_n)|n∈π_e(G) , s_n ∈ τ_e(G)\rbrace$, where $s_n$ is the number of elements with order $n$. Suppose that $T(G)=T(H)$. If $G$ is a finite solvable Group, is it true that H is also necessarily solvable? So far, no one can solve this problem completely even give a counterexample. In 1986, someone studied the case of the simple Group $A5$, and he proved an interesting result using only $π_e(G)$, that is, a finite Group $G$ is isomorphic to $A5$ if and only if $π_e(G)$ = {1, 2, 3, 5}. Afterward, many simple Groups are discovered by the characterization using only the set of element orders. Comparing the sizes of elements of same order but disregarding the actual orders of elements in $T(G)$ of the Thompson Problem, in other words, it remains only $τ_e(G)$, whether can it characterize finite simple Groups? Up to now, There are some papers that have characterized a Group $G$ only in terms of the set $τ_e(G)$ and there are some papers that have characterized $G$ by $τ_e(G)$ and some more conditions. Has the Thompson Problem been solved completely so far?