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Note on a paper of Tsuzuku
Published online by Cambridge University Press: 18 May 2009
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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.
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- Copyright © Glasgow Mathematical Journal Trust 1964
References
REFERENCES
1.Farahat, H. K., On the natural representation of the symmetric groups, Proc. Glasgow Math. Assoc. 5 (1962), 121–136.CrossRefGoogle Scholar
2.Tsuzuku, T., On decompositions of the permutation representation of a permutation group, Nagoya Math. J. 22 (1963), 79–82.CrossRefGoogle Scholar
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