# Internal free product

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

## Contents

## Definition (for two groups)

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

Note that this in particular implies that and intersect trivially.

The subgroups and are termed free factors of .

### Equivalence with external free product

`Further information: Equivalence of internal and external free product`

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

## Definition (for infinitely many groups)

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

Note that this in particular implies that each has trivial intersection with the subgroup generated by all the s, . 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.