Note on a paper of Tsuzuku
Glasgow mathematical journal, Tome 6 (1964) no. 4, pp. 196-197
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Farahat, H. K. Note on a paper of Tsuzuku. Glasgow mathematical journal, Tome 6 (1964) no. 4, pp. 196-197. doi: 10.1017/S2040618500035012
@article{10_1017_S2040618500035012,
author = {Farahat, H. K.},
title = {Note on a paper of {Tsuzuku}},
journal = {Glasgow mathematical journal},
pages = {196--197},
year = {1964},
volume = {6},
number = {4},
doi = {10.1017/S2040618500035012},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S2040618500035012/}
}
[1] 1.Farahat, H. K., On the natural representation of the symmetric groups, Proc. Glasgow Math. Assoc. 5 (1962), 121–136. Google Scholar | DOI
[2] 2.Tsuzuku, T., On decompositions of the permutation representation of a permutation group, Nagoya Math. J. 22 (1963), 79–82. Google Scholar | DOI
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