External free product

Definition in terms of words
Suppose $$G_1$$ and $$G_2$$ are groups. Let $$S$$ be the disjoint union of the non-identity elements of $$G_1$$ and $$G_2$$.

The external free product of $$G_1$$ and $$G_2$$, denoted $$G_1 * G_2$$ is defined as the set of words of finite length with letters from $$S$$, including the empty word, where the letters of the word alternate between $$G_1$$ and $$G_2$$, and where the multiplication is defined as follows:


 * If either word is empty, the product is simply the other word.
 * If $$a$$ and $$b$$ are two words such that the last letter of $$a$$ and the first letter of $$b$$ are from different groups, then the product is obtained simply by concatenating the words, i.e., writing $$a$$ followed by $$b$$.
 * If the last letter of $$a$$ (say $$g$$) and the first letter of $$b$$ (say $$h$$) are from the same group, we do the following. We concatenate the words, then we replace the two letters $$gh$$ by their product $$gh$$. If this product is not the identity element, we have the new word. If the product is the identity element, we delete the letter, and now replace the letter before $$g$$ and the letter after $$h$$ by their product. If this is the identity element, we again delete it. We keep iterating this process.

Associativity is easy to verify; the inverse of a word is simply the word with the inverses of the individual letters written in opposite order, and the identity element is the empty word.

Note here that when we consider the external free product of a group with itself, we are really considering the external free product of two disjoint copies of the group (i.e., disjoint except at the identity).

Equivalence with internal free product
If a group is the internal free product of two subgroups, it is naturally isomorphic to their external free product, where these subgroups are identified with the words of length one from those same subgroups.

Note that unlike the case of direct products, there is no distinction between restricted and unrestricted external free products.

Definition in terms of words
Suppose $$I$$ is an indexing set and $$G_i, i \in I$$ is a collection of groups. Let $$S$$ be the pairwise disjoint union of the non-identity elements of $$G_i$$. The external free product of the $$G_i$$s is defined as the group whose elements are words of finite length with letters from $$S$$, including the empty word, where no two adjacent letters of a word are from the same $$G_i$$, and where multiplication is defined as follows:


 * If either word is empty, the product is simply the other word.
 * If $$a$$ and $$b$$ are two words such that the last letter of $$a$$ and the first letter of $$b$$ are from different groups, then the product is obtained simply by concatenating the words, i.e., writing $$a$$ followed by $$b$$.
 * If the last letter of $$a$$ (say $$g$$) and the first letter of $$b$$ (say $$h$$) are from the same group, we do the following. We concatenate the words, then we replace the two letters $$gh$$ by their product $$gh$$. Then, if we find again that the last letter before the point of concatenation and the first letter after the point of concatenation are from the same group, we repeat the process.

Weaker product notions

 * External regular product

Commutativity
The external free product of $$G_1$$ and $$G_2$$ equals the external free product of $$G_2$$ and $$G_1$$. More generally, the external free product of any collection of groups does not depend on any ordering of that collection.

Associativity
The external free product is associative up to natural isomorphism:

$$G_1 * (G_2 * G_3) \cong (G_1 * G_2) * G_3$$.

Both of these are isomorphic to the external free product of the collection $$\{ G_1, G_2, G_3 \}$$ as defined in the sense of the external free product of a collection of groups.

More generally, the order of parenthesization for external free products does not matter, even in the case of infinitely many groups.

Identity element
The trivial group is the identity element for the free product, in the sense that the free product of any group with the trivial group is isomorphic to the original group.

Facts

 * Free product of nontrivial groups is infinite
 * Free product of nontrivial groups is centerless