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
- 1 History
- 2 Statement
- 3 Definitions
- 4 Justification/appeal of the conjecture
- 5 Progress towards the conjecture
- 6 References
- 7 External links
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.
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
Primitive group action
Character of a permutation representation
Justification/appeal of the conjecture
Progress towards the conjecture
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.
- Kourovka notebook: unsolved problems in group theory
- Google Print link for the preview of the Kourovka notebook