Let H1,H2 be subgroups of a finite group G. Assume that G = umlH2yiH1 = n Uj=1 HlgiH1 and that y1=1,g1= 1. Let Di be the set consisting of right cosets of H~ contained in H2yiH1 and let dj （j --- 1,..., n） be the s...Let H1,H2 be subgroups of a finite group G. Assume that G = umlH2yiH1 = n Uj=1 HlgiH1 and that y1=1,g1= 1. Let Di be the set consisting of right cosets of H~ contained in H2yiH1 and let dj （j --- 1,..., n） be the set consisting of right cosets contained in HlgjH1. We define the n x m matrix Mz （z = 1,..., m） whose columns and rows are indexed by Di and dj respectively and the （dk, Dz） entry is ｜Dzgk N Dl]. Let M =（M1 , Mm）. Assume that 1GH1 and 1H2G are semisimple permutation modules of a finite group G. In this paper, by using the matrix M, we give some sufficient and necessary conditions such that 1~1 is isomorphic to a submodule of 1H2G. As an application, we prove Foulkes＇ conjecture in special cases.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.10401034)
文摘Let H1,H2 be subgroups of a finite group G. Assume that G = umlH2yiH1 = n Uj=1 HlgiH1 and that y1=1,g1= 1. Let Di be the set consisting of right cosets of H~ contained in H2yiH1 and let dj （j --- 1,..., n） be the set consisting of right cosets contained in HlgjH1. We define the n x m matrix Mz （z = 1,..., m） whose columns and rows are indexed by Di and dj respectively and the （dk, Dz） entry is ｜Dzgk N Dl]. Let M =（M1 , Mm）. Assume that 1GH1 and 1H2G are semisimple permutation modules of a finite group G. In this paper, by using the matrix M, we give some sufficient and necessary conditions such that 1~1 is isomorphic to a submodule of 1H2G. As an application, we prove Foulkes＇ conjecture in special cases.
