Modular Representations of Finite Groups (Pure and Applied by B. M. Puttaswamaiah, John D. Dixon

By B. M. Puttaswamaiah, John D. Dixon

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Let U and V be KG-modules. Then show that i(U, V) is equal to the dimension of the space of invariants of U* B KV, where U*denotes the dual of u. 3. Let G be a group and K a splitting field for G. Show that a completely reducible KG-module is irreducible iff i(U, V )= 1. 4. Let G be a group, K a field, and U = e ( K G )a principal indecomposable KG-module generated by the idempotent e. Show that a KG-module V has a composition factor isomorphic to U/U'iff Ve # 0. 5. Let K be a field, U1,U 2, . If ).

We have to prove D = K. Suppose D # K, and choose a E D\K. are not linearly independent over K. Thus there exists a nonzero polynomialf(x) in K [ x ] with leading coefficient 1 such that f ( a ) = 0. Let L be a splitting field off@) over K and write f ( x ) = (x - a,)(x - a 2 ) (x - a,,,). Then in D B K L we have ( a @ 1 - 1 6a , ) ( a B 1 - 1 0 a 2 ) ( a 6 1 - 1 a,,,) = 0 with e . 1 a B 1 # 1 0 ai for any i because a $ K. But this shows that D O KL contains nonzero divisors of 0 and so is not a division ring.

V @ y l } and under this action the stabilizer of V @ y , I {z E L V @ x , z = V @ x l }= (z E LI = V@ x1 x 1 z x ; ' E H}= x ; ' H x , n L. 1 we conclude that = ( Vx,)" for each i, and the theorem is proved. W,z (V',)". Similarly M( Our next result also relates the induction and restriction processes. 2B Let A be a commutative ring, G a group, and H a subgroup of G. Let U be an AH-module and Van AG-module, and suppose that U and V are both free as A-modules (in particular, the latter always holds if A is a field).

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