Mordell’s theorem

Abstract

An abelian group is finitely generated if there exist finitely many elements a1, a2, . . . , ak such that any element of G can be expressed as a sum c1a1 + c2a2 + . . .+ ckak, where the ci are integers and multiplication denotes repeated addition. Note that this representation need not be unique, so any finite group is also finitely generated. A subgroup of an abelian group G is a set H ⊆ G which is itself a group under the same operation. For any a ∈ G, a+H = {a+ h : h ∈ H} is a coset of H. a is called a representative of the coset a + H. If b ∈ a + H, then b − a ∈ H. Any two cosets of H are either equal or disjoint. The index of H in G, denoted [G : H], is the number of disjoint cosets of H. For a ∈ G, the order of a is the minimum positive integer k such that ka is the identity, or ∞ if there is no such k.