Internal free product

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
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.

Weaker product notions

 * Stronger than::Internal regular product