Final answer:
Z x Z / <(1, 2)> is the quotient group formed by the abelian group Z x Z and its subgroup generated by (1, 2). By showing that every element in the quotient group can be represented with the first component being 0, we proved that the group is isomorphic to Z, which means the quotient group has the same structure as the group of all integers.
Step-by-step explanation:
We are working with the group Z × Z, which is an abelian group under addition. The subgroup <(1, 2)> is the set of all integer multiples of the element (1,2), which is a subgroup of Z × Z. To classify Z × Z / <(1, 2)> according to the fundamental theorem of finitely generated abelian groups, we need to understand that we're looking for a simpler description of this quotient group.
Now, we will consider an arbitrary element (m, n) + <(1, 2)> in the quotient group. We can manipulate this element to show that it is equivalent to an element where the first component is 0. Every element in Z × Z can be written as (0, n) + m*(1, 2) for some integer m. After adding the subgroup, this is (0, n) + <(1, 2)> = (0, n - 2m) + <(1, 2)> because subtracting a multiple of the subgroup does not change the equivalence class.
Since any element in the quotient group can be represented with the first component being 0, the quotient group is isomorphic to Z, which is the group of all integers under addition. The subgroup <(1, 2)> acts as a 'dimension reduction' from Z × Z to Z, as it allows us to 'ignore' the contribution of the first component. Therefore, the given group Z × Z / <(1, 2)> is indeed classified as isomorphic to Z.