Character determines primitivity conjecture

History
This conjecture was made by Wielandt at the Sixth All-Unio Symposium on Group Theory, held in Cherkassy in 1978.

In terms of permutation representations
Given two permutation representations (viz group actions) of a finite group, both having the same character, one of them is primitive if and only if the other is. In other words, the character of a permutation representation determines whether or not it is primitive.

In terms of subgroups of the symmetric group
Let $$P$$ and $$Q$$ be two isomorphic subgroups of the symmetric group $$Sym(n)$$ on $$n$$ letters, such that for every conjugacy class $$C$$ of $$Sym(n)$$, $$|P \cap C| = |Q \cap C|$$. Then, $$P$$ is a maximal subgroup of $$Sym(n)$$ if and only if $$Q$$ is.

Solvable groups
The conjecture has been settled for the case of solvable groups. In other words, if $$G$$ is a solvable group, and there are two permutation representations of $$G$$ with the same character, one of them is primitive if and only if the other is.