Exercise 3, § 21

Let A be a commutative ring and G a group.

a) Suppose that the algebra A[G] of the group G is semisimple. Prove that A is semisimple, that G is finite, and that its order is invertible in A (use Exercise 2 and Exercise 25 of VIII, p. 22).

b) Suppose that the algebra A[G] is isomorphic to the product of a finite family of fields. Prove that every subgroup of G is normal (use a), and observe that every idempotent in A[G] is central).

In Exercises 4 through 26 below, G is a finite group, and K is an algebraically closed field; we assume that the order of G is invertible in K.