Character determines primitivity conjecture
This article is about a conjecture in the following area in/related to group theory: permutation groups. View all conjectures and open problems
This conjecture is believed to be true
Contents
History
This conjecture was made by: Wielandt
This conjecture was made by Wielandt at the Sixth All-Unio Symposium on Group Theory, held in Cherkassy in 1978.
Statement
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 and be two isomorphic subgroups of the symmetric group on letters, such that for every conjugacy class of , . Then, is a maximal subgroup of if and only if is.
Definitions
Primitive group action
Character of a permutation representation
Justification/appeal of the conjecture
Progress towards the conjecture
Solvable groups
The conjecture has been settled for the case of solvable groups. In other words, if is a solvable group, and there are two permutation representations of with the same character, one of them is primitive if and only if the other is.
References
- Kourovka notebook: unsolved problems in group theory
External links
- Google Print link for the preview of the Kourovka notebook