# Internal free product

This article describes a product notion for groups. See other related product notions for groups.

## Definition (for two groups)

Suppose $G_1$ and $G_2$ are subgroups of a group $G$. $G$ is termed an internal free product of $G_1$ and $G_2$ if for any non-empty word whose letters alternate between the non-identity elements of $G_1$ and $G_2$, the corresponding element of $G$ is not the identity element. In other words, there is no nontrivial relation between $G_1$ and $G_2$.

Note that this in particular implies that $G_1$ and $G_2$ intersect trivially.

The subgroups $G_1$ and $G_2$ are termed free factors of $G$.

### Equivalence with external free product

Further information: Equivalence of internal and external free product

If $G$ is the internal free product of subgroups $G_1$ and $G_2$, it is naturally isomorphic to the external free product of $G_1$ and $G_2$.

## Definition (for infinitely many groups)

Suppose $G_i, i \in I$ is a (possibly infinite) collection of subgroups of a group $G$. $G$ is termed an internal free product of the $G_i$s if for any non-empty word whose letters are drawn from the non-identity elements of $G_i$, with no two consecutive letters in the same $G_i$, the corresponding element of $G$ is not the identity element. In other words, there is no nontrivial relation between the $G_i$s.

Note that this in particular implies that each $G_i$ has trivial intersection with the subgroup generated by all the $G_j$s, $j \ne i$. Note also that if a subgroup is part of a (possibly infinite) free product, it is also part of a free product involving two subgroups, hence is a free factor.