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, $j\neq i$. 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.

Relation with other product notions

Weaker product notions

• Internal regular product