By Eschenburg J. H., Tribuzy R.

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**Example text**

4) THEOREM. The following are equivalent for a finite group G: (1) G is abelian, (2) all simple complex representations of G have degree 1. Proof. Since the simple representations of every finite cyclic group all have degree 1, and since every finite abelian group is a direct product of cyclic groups, the last result shows that all simple representations of a finite abelian group have degree 1. r Conversely, we may use the fact that |G| = i=1 d2i where d1 , . . , dr are the degrees of the simple representations.

1. Frobenius reciprocity is a most useful tool in calculating with induced characters. In the special case that V and W are simple representations over C of H and G, respectively, where H ≤ G, it says that the multiplicity of W as a summand of G V ↑G H equals the multiplicity of V as a summand of W ↓H . As an example we may take both V and W to be the trivial representations of their respective groups. 9, C ↑G H is a permutation module. We deduce from Frobenius reciprocity that as representations of G, C is a direct summand of C ↑G H with multiplicity one.

R be the simple complex characters of G with degrees d1 , . . , dr . The primitive central idempotent elements in CG are the elements di |G| χi (g −1 )g g∈G Finite Group Representations Printed Dec. 12, 2007 32 where 1 ≤ i ≤ r, the corresponding indecomposable ring summand of CG having a simple representation which affords the character χi . Proof. 20 we have that the representation ρi which affords χi yields an algebra map ρi : CG → Mdi (C) which is projection onto the ith matrix summand in a decomposition of CG as a sum of matrix rings.