Free Products of Groups

Suppose one is given a collection of groups \(G_\alpha\) and one wishes to construct a single group containing all these groups as subgroups. One way to do this would be to take the product group \(\prod_\alpha G_\alpha\), whose elements can be regarded as the functions \(\alpha \mapsto g_\alpha \in G_\alpha\). Or one could restrict to functions taking on nonidentity values at most finitely often, forming the direct sum \(\bigoplus_\alpha G_\alpha\). Both these constructions produce groups containing all the \(G_\alpha\)’s as subgroups, but with the property that elements of different subgroups \(G_\alpha\) commute with each other. In the realm of nonabelian groups this commutativity is unnatural, and so one would like a ‘nonabelian’ version of \(\prod_\alpha G_\alpha\) or \(\bigoplus_\alpha G_\alpha\). Since the sum \(\bigoplus_\alpha G_\alpha\) is smaller and presumably simpler than \(\prod_\alpha G_\alpha\), it should be easier to construct a nonabelian version of \(\bigoplus_\alpha G_\alpha\), and this is what the free product \({\Large *}_\alpha G_\alpha\) achieves.

Here is the precise definition. As a set, the free product \({\Large *}_\alpha G_\alpha\) consists of all words \(g_1g_2 \cdots g_m\) of arbitrary finite length \(m \geq 0\), where each letters \(g_i\) belongs to a group \(G_{\alpha_i}\) and is not the identity element of \(G_{\alpha_i}\), and the adjacent letters \(g_i\) and \(g_{i+1}\) belong to different groups \(G_\alpha\), that is, \(\alpha_i \neq \alpha_{i+1}\). Words satisfying these conditions are called reduced, the idea being that unreduced words can always be simplified to reduced words by writing adjacent letters that lie in the same \(G_{\alpha_i}\) as a single letter and by canceling trivial letters. The empty word is allowed, and will be identity element of \({\Large *}_\alpha G_\alpha\). The group operation in \({\Large *}_\alpha G_\alpha\) is juxtaposition, \((g_1 \cdots g_m)(h_1 \cdots h_n) =\) :math:` g_1 cdots g_m h_1 cdots h_n`. This product may not be reduced, however: If \(g_m\) and \(h_1\) belong to the same \(G_\alpha\), they should be combined into a single letter \((g_mh_1)\) according to the multiplication in \(G_\alpha\), and if this new letter \(g_mh_1\) happens to be the identity of \(G_\alpha\), it should be canceled from the product. This may allow \(g_{m-1}\) and \(g_2\) to be combined, and possibly canceled too. Repetition of this process eventually produces a reduced word. For example, in the product \((g_1 \cdots g_m)(g^{-1}_m \cdots g^{-1}_1)\) everything cancels and we get the identity element of \({\Large *}_\alpha G_\alpha\), the empty word.

Verifying directly that this multiplication is associative would be rather tedious, but there is an indirect approach that avoids most of the word. Let \(W\) be the set of reduced words \(g_1 \cdots g_m\) as above, including the empty word. To each \(g \in G_\alpha\) we associate the function \(L_g : W\rightarrow W\) given by multiplication on the left, \(L_g(g_1 \cdots g_m)=\) \(gg_1 \cdots g_m\) where we combine \(g\) with \(g_1\) if \(g_1 \in G_\alpha\) to make \(gg_1 \cdots g_m\) a reduced word. A key property of the association \(g \mapsto L_g\) is the formula \(L_{gg'}=L_gL_{g'}\) for \(g,g' \in G_\alpha\), that is, \(g(g'(g_1 \cdots g_m)) = (gg')(g_1 \cdots g_m)\). This special case of associativity follows rather trivially from associativity in \(G_\alpha\). The formula \(L_{gg'} = L_gL_{g'}\) implies that \(L_g\) is invertible with inverse \(L_{g^{-1}}\). Therefore the association \(g \mapsto L_g\) defines a homomorphism from \(G_\alpha\) to the group \(P(W)\) of all permutations of \(W\). More generally, we can define \(L:W \rightarrow P(W)\) by \(L(g_1 \cdots g_m) = L_{g_1} \cdots L_{g_m}\) for each reduced word \(g_1 \cdots g_m\). This function \(L\) is injective since the permutation \(L(g_1 \cdots g_m)\) sends the empty word to \(g_1 \cdots g_m\). The product operation in \(W\) corresponds under \(L\) to composition in \(P(W)\), because of the relation \(L_{gg'}=L_gL_{g'}\). Since composition in \(P(W)\) is associative, we conclude that the product in \(W\) is associative.

In particular, we have the free product \(\mathbb{Z} * \mathbb{Z}\) as described earlier. This is an example of a free group, the free product of any numbers of copies of \(\mathbb{Z}\), finite or infinite. The elements of a free group are uniquely representable as reduced words in powers of generators for the various copies of \(\mathbb{Z}\), with one generator for each \(\mathbb{Z}\), just as in the case of \(\mathbb{Z} * \mathbb{Z}\). These generators are called a basis for the free group, and the number of basis elements is the rank of the free group. The abelianization of a free group is a free abelian group with basis the same set of generators, so since the rank of a free abelian group is well-defined, independent of the choice of basis, the same is true for the rank of a free group.

An interesting example of a free product that is not a free group is \(\mathbb{Z}_2 * \mathbb{Z}_2\). This is like \(\mathbb{Z} * \mathbb{Z}\) but simpler since \(a^2 = e = b^2\), so powers of \(a\) and \(b\) are not needed, and \(\mathbb{Z}_2 * \mathbb{Z}_2\) consists of just the alternating words in \(a\) and \(b\): \(a\), \(b\), \(ab\), \(ba\), \(aba\), \(bab\), \(abab\), \(baba\), \(ababa\), \(\cdots\), together with the empty word. The structure of \(\mathbb{Z}_2 * \mathbb{Z}_2\) can be elucidated by looking at the homomorphism \(\varphi : \mathbb{Z}_2 * \mathbb{Z}_2 \rightarrow \mathbb{Z}_2\) associating to each word its length mod \(2\). Obviously \(\varphi\) is surjective, and its kernel consists of the words of even length. These form a infinite cyclic subgroup generated by \(ab\) since \(ba=(ab)^{-1}\) in \(\mathbb{Z}_2 * \mathbb{Z}_2\). in fact, \(\mathbb{Z}_2 * \mathbb{Z}_2\) is the semi-direct product of the subgroups \(\mathbb{Z}\) and \(\mathbb{Z}_2\) generated by \(ab\) and \(a\), with the conjugation relation \(a(ab)a^{-1}=(ab)^{-1}\). This group is sometimes called the infinite dihedral group.

For a general free product \({\Large *}_\alpha G_\alpha\), each group \(G_\alpha\) is naturally identified with a subgroup of \({\Large *}_\alpha G_\alpha\), the subgroup consisting of the empty word and the nonidentity one-letter words \(g \in G_\alpha\). From this viewpoint the empty word is the common identity element of all the subgroups \(G_\alpha\), which are otherwise disjoint. A consequence of associativity is that anay product \(g_1 \cdots g_m\) of elements \(g_i\) in the groups \(G_\alpha\) has a unique reduced form, the element of \({\Large *}_\alpha G_\alpha\) obtained by performing the multiplications in any order. Any sequence of reduction operations on an unreduced product \(g_1 \cdots g_m\), ombining adjacent letters \(g_i\) and \(g_{i+1}\) that lie in the same \(G_\alpha\) or canceling a \(g_i\) that is the identity, can be viewed as a way of inserting parentheses into \(g_1 \cdots g_m\) and performing the resulting sequence of multiplications. Thus associativity implies that any two sequences of reduction operations performed on the same unreduced word always yield the same reduced word.

A basic property of the free product \({\Large *}_\alpha G_\alpha\) is that any collection of homomorphisms \(\varphi_\alpha : G_\alpha \rightarrow H\) extends uniquely to a homomorphism \(\varphi : {\Large *}_\alpha G_\alpha \rightarrow H\). Namely, the value of \(\varphi\) on a word \(g_1 \cdots g_n\) with \(g_i \in G_{\alpha_i}\) must be \(\varphi_{\alpha_1}(g_1) \cdots \varphi_{\alpha_n}(g_n)\), and using this formula to define \(\varphi\) gives a well-defined homomorphism since the process of reducing an unreduced product in \({\Large *}_\alpha G_\alpha\) does not affect its image under \(\varphi\). For example, for a free product \(G {\Large *} H\) the inclusions \(G \hookrightarrow G \times H\) and \(H \hookrightarrow G \times H\) induce a surjective homomorphism \(G {\Large *} H \rightarrow G \times H\).