Question

Prove the Universal Mapping Property for Z×Z : If g and h are any elements of a group G, then there exists a unique homomorphism ϕ:Z×Z⟶G such that ϕ(1,0)=g and ϕ(0,1)=h if and only if gh=hg.

Answer 1

To prove the Universal Mapping Property for Z×Z, we need to show that for any two elements g and h in a group G, there exists a unique homomorphism ϕ: Z×Z ⟶ G such that ϕ(1,0) = g and ϕ(0,1) = h if and only if gh = hg. We can define the homomorphism ϕ as ϕ(a, b) = a·g + b·h, where a and b are integers.

Step-by-step explanation:

To prove the Universal Mapping Property for Z×Z, we need to show that for any two elements g and h in a group G, there exists a unique homomorphism ϕ: Z×Z ⟶ G such that ϕ(1,0) = g and ϕ(0,1) = h if and only if gh = hg.

To prove this, we can define the homomorphism ϕ as follows:

ϕ(a, b) = a·g + b·h

where a and b are integers. Now, let's prove the two conditions of the Universal Mapping Property:

1. ϕ(1, 0) = g:
2. ϕ(1, 0) = 1·g + 0·h = g
3. ϕ(0, 1) = h:
4. ϕ(0, 1) = 0·g + 1·h = h

Thus, the homomorphism ϕ satisfies both conditions of the Universal Mapping Property, and gh = hg holds.