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