I have been self-studying from Aluffi's Algebra: Chapter 0. I am looking at Chapter II section 3, exercise 3.8. Here it is:
As far as I can tell, I need to do the following:
- Describe the projections $\pi_2 : C_2 \to G$ and $\pi_3 : C_3 \to G$.
- Show that the universal property for coproducts is satisfied: that is, for any group $A$ and any choice of group homomorphisms $\varphi_2 : C_2 \to A$ and $\varphi_3 : C_3 \to A$, there exists a unique group homomorphism $\sigma : G \to A$.
Giving a more concrete description of the cyclic groups, take $C_2 = \{0, 1\}$ and $C_3 = \{0, 1, 2\}$. Then the projections are $\pi_2(k) = x^k$ where $k=0,1$, and $\pi_3(l) = y^l$ where $l=0,1,2$. I think that this takes care of (1).
Beyond this, it is not clear to me what I should do. Any help would be appreciated!
Best Answer
You can look at $G$ as a group that has strings like $xyxy^2xy$ or $y^2xyxy$ as elements.
Then $\phi:G\to A$ on e.g. $xyxy^2xy$ must be prescribed by:$$xyxy^2xy\mapsto\phi_2(x)\phi_3(y)\phi_2(x)\phi_3(y)^2\phi_2(x)\phi_3(y)$$
This $\phi$ must be shown to be a group homomorphism with $\phi\circ\pi_2=\phi_2$ and $\phi\circ\pi_3=\phi_3$, and must be shown to be unique in satisfying this.