The Unapologetic Mathematician

Mathematics for the interested outsider

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.


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


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

    Comment by edriv | December 11, 2007 | Reply

  2. 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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