By Murakami M.

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**Example text**

FC1CtOI' of' A. 7 Theorem. Let A and U be finite abelian groups. Theifthe number of B :::; A Proof. a) Let 62 with B ~ U and the number of C :::; A with AIC ::::: U are equal. Proof. As for B :::; A we have Bll. 6a), so B --+ Bl is injective. If D:::; A, then Dll = D, hence B --+ Bl is bijective. As AIBl ~ fj ~ B, so AIBl. ~ U is equivalent to B ~ U. As A ~ A, we obtain the theorem. d. 8 Proposition. ) a) Um(A)l. = Dm(A). b) Dm(A)l. = Um(A). b) We claim that Exp A = Exp A: If), E A and a E A, then ),IIl(a) = ),(Olll) = 1, hence }1Il = 1.

If bE B, then til Irr G such that E (i, lj;)c = ° for i = 0, ... , k - 1. Then °= I k X(g)ilj;(g-l) = for all 11. Choose In X(b)2 Therefore either X(b) = - ----------- I If;(g-l). gEAj ° (j = 1, ... , k). 18c) = lj;(g-l) = 9 E Aj ° Al = {glx(g) = x(1)} = Ker X = E. Z(B). b) ~ a): Suppose now G = A x B as above. 2 = on B\Z(B). Hence ° I 1. Then = X(1)2 m-1 X(b 2m ) = X(I)2 m -l X(b). 2 aj As the Vandermonde determinant det(aj) is different from 0, this system of homogeneous linear equations has only the trivial solution, hence such that ord bl2 m m I )=1 gEC A trivial induction shows (i=1, ...

B) If W is a cyclic CG-module of the form W = wCG, then a with aa = \Va for a E CG is a CG-epimorphism of CG onto W. Hence cyclic CG-modules are just the epimorphic images of CG. 7 in CG with multiplicity Xi(1). d. CG-module if and only if In i ::::; X;(l). Using a), we see that b) is true. 7 Theorem. Let V be a CG-l11odule with character X. 011 V 09c V we define a C-linear mapping T by a) Then T E HomcG(V@c V, V 09c V), and V 09c V = A ffi S, Products of representations and characters Products of representations and characters 100 101 As g2 has the eigenvalues a i (g)2 on V, we obtain where A = Ker(t + 1) and S = Ker(t - 1) (5(g) are CC-modules.