# Universal pair proofs

Definition:

A pair (U, epsilon) is universal for a group G, with respect to abelian homomorphic images, if U is an abelian group, epsilon:G --> U, an epimorphism, such that for any other abelian group A and surjection f:G --> A, there exists a unique g:U --> A such that f=g composed with epsilon. In this case, we say f can be factored through U.

This is the question:

If (U, epsilon) is a universal pair for a group G and h is in Aut(U), show that (U, h*epsilon) is also universal for G.

If (U, epsilon_1) is universal for G, show that epsilon_1=h*epsilon for some h in Aut(U)

I need a detailed proof of this to study please.

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#### Solution Preview

(1) Suppose (U, e) is universal pair for G, and let h be an automorphism of U and f : G --> A a surjection onto any abelian

group A; then he : G --> U is epi, and f = (gh^{-1})(he) where g is the unique ...

#### Solution Summary

This provides examples of proofs with universal pairs.