Group algebras, decompositions question...

Hi everyone,



I am trying to do the following problem. Let F be a field of order q, G a
finite abelian group of order n, such that q,n are relatively prime. I want
to show that the group algebra FG is isomorphic to a direct sum of fields
E_1 (+) ... (+) E_t where E_i is a field with q^(e_i) elements, with n = e_1
+ e_2 + ... + e_t.



I was wondering, if G and H are group and R is a ring, is



R(G (+) H) isomorphic to RG (+) RH? I can't seem to find a nice map that
gives me an isomorphism.



(Where I'm talking about group rings here)



Well by Maschke's theorem, FG is semisimple so it's the direct sum of simple
rings (fields?)



I think I know what QG is when G is a cyclic group of just prime order (not
of prime power order) and Q is the rationals, but when I have a finite
abelian group (the direct product of cyclic groups), I'm not sure how to do
this..and it would be very nice if the property I asked about holds for
group rings and direct sums.

Please let me know if I'm thinking about this correctly : Let G = Z/pZ.
Then QG = Q Z/pZ = Q[x]/(x^p-1) = Q[x]/(x-1) (+) Q[x]/(x^(p-1) + ... + x +
1) = Q (+) some field.

Any help?

Thanks in advance,

Tony


.

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