free product (groups)

Free product

This is a way to make a group from other groups that is similar to direct product but is a way to “fix” how the commuting property of the direct product. i.e.

$$(g,e)\cdot (e,h)=(g,h)=(e,h)\cdot (g,e)\forall g\in G,h\in H$$

Definition

For a collection of groups $\{G_i{\}}_{i\in I}$, the free product is a group

$${\prod _{i\in I}}^{\ast }{G}_i$$

together with group homomorphism

$${\varphi }_i:G_i\hookrightarrow {\prod _{i\in I}}^{\ast }{G}_i$$

such that for any group H and a homomorphism

$${\psi }_i:G_i\to H$$

There exists a unique homomorphism

$$\psi :{\prod _{i\in I}}^{\ast }{G}_i\to H$$

such that

Proof of Existence

todo

Example

$\cdots a^{♯}{b}^{♯}{a}^{♯}\cdots \in \mathbb{Z}\ast \mathbb{Z}$ Then for an example of group multiplication we have $a^5{b}^3{a}^{-1}\cdot a^2{b}^3=a^5{b}^{3}a{b}^3$