Character determines primitivity conjecture

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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

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 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.

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 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.

References

  • Kourovka notebook: unsolved problems in group theory

External links

  • Google Print link for the preview of the Kourovka notebook