Internal free product

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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_is 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_is.

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

Relation with other product notions

Weaker product notions