Let be a field and a positive integer. Prove that is normal in and describe the isomorphism type of .
- Let and . Now , since multiplication in is commutative. Thus . Hence is normal in .
- Define a mapping by .
- (Well-defined) Suppose such that . Then , so that . Thus is well defined.
- (Homomorphism) We have . Thus is a homomorphism.
- (Injective) Suppose . Then , and we have , so that . Hence , and is injective.
- (Surjective) For all , note that the matrix with in the entry, 1 in all other diagonal entries, and 0 in all off diagonal entries has determinant . Thus is surjective.
Thus is a group isomorphism, so that .