# Groups and Uniqueness of Decomposition

Let A = Z + Z + Z, let x_1 = (1,2,1) and x_2 = (1,5,1), and consider the subgroup B = (x_1, x_2) is a member of G. For the quotient group G = A/B, and write (x,y,z) is a member of G for the coset determined by n element (x,y,z) is a member of A.

a) For the subgroups J_1 = {x,y,0 for | x,y is a member of Z} and J_2 = {(0,0,z) | z is a member of Z} is a member of G. Show that G is now the internal direct sum J_1 + J_2.

b) For the subgroups H_1 = {x,y,x for | x,y is a member of Z} and H_2 = {(0,0,z) | z is a member of Z} is a member of G. Show that G is now the internal direct sum H_1 + H_2.

c) Show that H_2 ~ Z.

d) Show that H_1 ~ Z/3Z.

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#### Solution Preview

(a)

Any element of G can be a sum of elements of J_1 and J_2, however the definition of direct sum (see http://en.wikipedia.org/wiki/Direct_sum) involves uniqueness of decomposition g = j_1 (+) j_2.

This uniqueness is violated by J_1 and J_2, as we can see from the following example:

bar(1,2,0) and bar(2,4,0) are distinct co-sets and so are different elements of J_1

bar(0,0,1) and bar (0,0,2) are district co-sets and so are different elements of J_2

These co-sets can be combined in two different ways to ...

#### Solution Summary

This solution shows how to solve for the given subgroups by using a step by step methodology. Explanations also accompany the steps. A reference which is used for a part of this solution is also given.