# The Unapologetic Mathematician

## 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 -