What is a free product of groups or vector space?

[Heidi]

Hi Pfs,
I do not succeed to handle free products of groups or vector spaces.
In the case of two vector spaces E and F the product (E,F) is the same thing that the free product E * F
I rad this article
https://en.wikipedia.org/wiki/Free_product
i would like to construct a free product in simple cases (say with groups of 2*2 matrices or somehthing
like that)
thanks

 

[martinbn]

What exactly is yout question?

 

[nuuskur]

The free product of two nontrivial groups is infinite. It's difficult to exhibit examples other than describe the generating process, which is outlined in your link, already.

It also doesn't matter how one labels the elements in the groups. For instance, we can have matrices $A,B,C$ and permutations $\sigma,\rho,\tau$. In the free product we just have words that might look something like $A\sigma B\rho\tau C$ and so on. There is nothing about matrices or mappings that stands out here.