The Indecomposables of Rank 3 Permutation Modules

Citation

Lewy, Michael Robert (1985) The Indecomposables of Rank 3 Permutation Modules. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6h5n-b393. https://resolver.caltech.edu/CaltechTHESIS:01222019-125147510

Abstract

Transitive permutation groups of finite order are viewed as linear groups over fields of characteristic p > 0 by having the group permute the basis elemerits of a vector space M. The decomposition of M into the direct sum of invariant subspaces is investigated, and criteria given for whether M is decomposable, and if it is, how many direct summands occur, in the special case the group has rank 3, i.e., it has 3 orbits on ordered pairs of points. In the case that each orbit is self-paired, M decomposes into the maximum possible number of indecomposables, and the group has every p'-element conjugate to its inverse, irreducibility results are obtained for the indecomposables. This last result holds for any rank. It applies in particular to the symmetric and thence to the alternating groups, which enables us to describe certain modular irreducibles of these groups.

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