---
tag: TM7QS7P7X9
kind: concept
name: Free product
msc: ["20E06"]
sources:
  - url: https://en.wikipedia.org/wiki/Free_product
same_as:
  wikidata: Q1454165
  wikipedia: Free product
---

## Summary

The free product $G * H$ of two [[groups|G5S1M9]] is the group generated by $G$ and $H$ with no relations between them.
Its elements are reduced words that alternate between non-identity elements of $G$ and of $H$.
It is the coproduct in the category of groups.
The free group of rank $n$ is the free product of $n$ copies of $\Z$.

## Explanations

### L1 human

To form a free product, put two groups side by side and let their elements multiply freely, without imposing any commutation between them.
An element is a word such as $g_1 h_1 g_2 h_2$, with letters taken alternately from the two groups.
For example, $\Z/2 * \Z/2$ is generated by two reflections $s$ and $t$, and the words $st$, $stst$, $ststst$ are all different, so the group is infinite.
In topology, the fundamental group of two well-behaved spaces, such as graphs, glued at a point is the free product of their fundamental groups.