A difficult exercise

While proctoring the make-up exam for my class, I thought of an exercise related to my group theory posts on direct and free products that should cause even the more experienced mathematicians in the audience a bit of difficulty.

Consider four groups $A_1$ , $A_2$ , $B_1$ , and $B_2$ , and four homomorphisms:

$f_{1,1}:A_1\rightarrow B_1$
$f_{1,2}:A_1\rightarrow B_2$
$f_{2,1}:A_2\rightarrow B_1$
$f_{2,2}:A_2\rightarrow B_2$

Use these to construct two homomorphisms from $A_1*A_2$ to $B_1\times B_2$ , and show that they’re the same.

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March 1, 2007 - Posted by John Armstrong | Algebra, Group Examples, Group theory, Universal Properties

2 Comments »

Is it time for a hint?
Where to find the second homomorphism?

Comment by edriv | December 11, 2007 | Reply
One comes from each universal property. One from the product and one from the coproduct.

Comment by John Armstrong | December 11, 2007 | Reply

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