1. Note on a paper of Tsuzuku
- Author
-
H. K. Farahat
- Subjects
Combinatorics ,Symmetric group ,Applied Mathematics ,General Mathematics ,Matrix representation ,Field (mathematics) ,Commutative ring ,Permutation matrix ,Permutation group ,Element (category theory) ,Unit (ring theory) ,Mathematics - Abstract
In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.
- Published
- 1964