# The free product

The coproduct in the category of groups is the free product $${G*H}$$, defined as the set of finite ordered “words” $${g_{1}h_{2}g_{3}h_{4}\dotsm g_{n-1}h_{n}}$$ of non-identity elements, with the group operation applied as juxtaposition of words. After juxtaposition, any adjacent letters from the same group are combined, and removed if the result is the identity. Consistent with this, the free product of a family of groups $${G_{\mu}}$$, denoted $${*G_{\mu}}$$, is defined as finite words with the letters being non-identity elements from any $${G_{\mu}}$$, where no adjacent elements come from the same $${G_{\mu}}$$.

 Δ It is important to note the difference between the free product $${*G_{\mu}}$$, where each letter $${g_{i}}$$ is an element of any $${G_{\mu}}$$ distinct from the previous one, and the direct sum $${\bigoplus G_{\mu}}$$, which can be viewed as a word where each letter $${g_{\mu}}$$ is an element of a distinct $${G_{\mu}}$$ (i.e. $${g_{\mu}}$$ is the only element of $${G_{\mu}}$$ in the word).

The free product is an example of the more general free object in category theory, which can be thought of as “forcing” one category into being another in the “most general” way; again we will not go into exact definitions, but instead describe some common free constructions.

• The free group on a set $${S}$$ “forces” $${S}$$ into being a group, defining inverses by a copy $${S^{\prime}}$$ and forming a group out of the finite ordered words of elements of $${S\cup S^{\prime}}$$ with juxtaposition as the group operation (as with the free product, any combinations $${ss^{-1}}$$ or $${s^{-1}s}$$ are removed)
• The free associative algebra on a vector space $${V}$$ (AKA the associative algebra $${W}$$ freely generated by $${V}$$) “forces” the words $${v_{1}v_{2}\dotsm v_{n}}$$ into being an associative algebra by defining vector multiplication as juxtaposition and requiring it to be multilinear, i.e. $${\left(v_{1}+v_{2}\right)\left(av_{3}\right)\equiv av_{1}v_{3}+av_{2}v_{3}}$$; as we will see below, this is in fact just the tensor algebra over $${V}$$, so an element $${v_{1}v_{2}}$$ can be written $${v_{1}\otimes v_{2}}$$
• The free module of rank $${n}$$ over a ring $${R}$$ has no multiplication, so the words of a specified length $${r_{1}r_{2}\dotsm r_{n}}$$ are “forced” into being a module by defining addition and multiplication component-wise, i.e. $${t\left(r_{1}r_{2}\dotsm r_{n}+s_{1}s_{2}\dotsm s_{n}\right)=\left(tr_{1}+ts_{1}\right)\left(tr_{2}+ts_{2}\right)\dotsm\left(tr_{n}+ts_{n}\right)}$$; thus an element $${r_{1}r_{2}}$$ is just a direct sum, and can be written $${r_{1}\oplus r_{2}}$$

The free abelian group is the free module of rank $${n}$$ over $${\mathbb{Z}}$$, since as we noted previously, any abelian group can be viewed as a module over $${\mathbb{Z}}$$ under “integer multiplication.” In fact, the free abelian group of rank $${n}$$ is just $${\mathbb{Z}\oplus\mathbb{Z}\oplus\dotsb\oplus\mathbb{Z}}$$ ($${n}$$ times) under component-wise addition.

 Δ Note that the name “free abelian group” is a potential source of confusion, since it is a free module, not a free group (except for the case of rank one, i.e. $${\mathbb{Z}}$$).