Prove
Lemma 52 ([Rank bound with composition length]). Let H<Y be connected classical…Theorem 6.1 ([Uniform absolute composition length]). In the setting of…Proposition
Use the same representation φ:Y(V) throughout the proof. For (Y,H,V) , every row of Table outside S_6 and MR_5 gives the conclusion. Suppose one of these two rows occurs. Then p=2 , the image of H is B_r , the image of Y is D_m or B_m , and V is a half-spin or spin module, respectively. Let W be the orthogonal natural module of this fixed ambient image, on a cover when necessary. Every nontrivial composition factor of W|_H has dimension 2r . Since H<X , irreducibility on H implies irreducibility on X . Apply the irreducible-triple Theorem a second time to (Y,X,V) , keeping the ambient group and representation unchanged. The theorem requires restrictedness and tensor-indecomposability on Y , so it makes no such assumption about the X -module V|_X . If the ambient image is B_m , the only row with a simple subgroup of rank at least ten is MR_5 . If it is D_m and V is half-spin, the only such rows are IV_1 , IV'_1 , IV'_2 , and S_6 . In all these cases the image of X is B_s and every nontrivial composition factor of W|_X is a Frobenius twist of N_X , of dimension 2s . By hypothesis and, each such constituent remains irreducible on H . Restricting a composition series for W|_X and refining it over H therefore exhibits an H -composition factor of dimension 2s . This contradicts the preceding dimension 2r , since s>r . The argument works for nonsplit natural restrictions as well as for direct sums. There is no extra case from MR_4 : its ambient image is C_m , whereas the image of the fixed representation here is B_m or D_m . One cannot change that image between the two applications. Equations-- preserve the natural irreducibility hypothesis on passing to these image groups, including an inseparable isogeny. They do not replace the fixed spin representation V by a natural representation. Finally, if X=Y , the irreducibility of N_Y|_H directly excludes S_6 and MR_5 , since their natural restrictions have at least two nontrivial composition factors.