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The main objective of the research is to study the first natural triangular representation of the symmetric groups over a field K of characteristic p≠ 2 which deals with the partition λ = (n − 4,3,1) of the positive integer n. Furthermore, this work has proven that the S(n − 4,3,1) is a submodule of F1 . The F1 = KSn(x2x4x6x7² −x2x3x6x7² − x2x4x5x7² + x²x³x5x7² ) can be only split whenp|(n − 5). الهدف من هذا البحث هو دراسة [ABSTRACT FROM AUTHOR]

In this work, we prove by employing mapping Cone that the sequence and the subsequence of the characteristic-zero are exact and subcomplex respectively in the case of partition (6,6,4) . [ABSTRACT FROM AUTHOR]