group-theory « Bedtime mathematics

associativity of amalgams or why universal properties can be powerful

October 23, 2009 mymbl 2 comments

Amalgams of groups are examples of what are known as colimits in category theory(aka abstract non-sense). With abstract non-sense, you can turn a blind-eye to the structure of the groups and concentrate wholly on its social behavior, determined by its universal property. There is nothing deep about proving results using universal properties of groups, as we are using the most obvious properties of groups when interacting with other groups. But these can be super-powerful when it comes to proving stuff which would otherwise be non-trivial ( for example when you are only looking at the structure of these groups).

Recall that the product of groups ${G_i}$ where ${i \in {\mathbb N}}$ , amalgamated with groups ${A_i}$ using injective homomorphisms ${f_i: A_i \rightarrow G_i}$ and ${f_{i}':A_i \rightarrow G_{i+1}}$ for ${i=1,2,\ldots}$ is the group ${U}$ along with homomorphisms ${g_i:G_i \rightarrow U}$ that are compatible with ${f_i}$ and ${f_i'}$ . This means that ${g_i \circ f_i = g_{i+1} \circ f_{i}'}$ for all ${i}$ . Further, ${U}$ has the universal property that if ${H}$ is any group with homomorphisms ${h_i: G_i \rightarrow H}$ which are compatible with the maps ${f_i,f_i'}$ for each ${i \in I}$ , we have a unique homomorphism ${\Phi : U \rightarrow H}$ such that:

$\displaystyle h_i = \Phi \circ g_i \text{ for } i \in \mathbb N$

The group ${U}$ is usually denoted by ${G_1 *_{A_1} G_2 *_{A_2} G_3 *_{A_3} \ldots}$ . To illustrate the power of this definition, lets us prove the following assertion about associativity of amalgams.

$\displaystyle (G_1 *_{A_1} G_2) *_{A_2} G_3 \equiv G_1 *_{A_2} G_2 *_{A_3} G_3$

Note that the amalgam ${G_1 *_{A_2} G_2 *_{A_3} G_3}$ is exactly the group that makes the following diagram commute and

whats more important: it is best candidate for making the diagram commute in the sense that, if ${H}$ is any other group replacing ${G_1 *_{A_1} G_2 *_{A_2} G_3}$ that also makes the above diagram commute then there is a unique homomorphism from ${G_1 *_{A_1} G_2 *_{A_2} G_3}$ to ${H}$ . Such a ${G_1 *_{A_1} G_2 *_{A_2} G_3}$ is hence unique upto a unique isomorphism. To prove the above equality we try to show that LHS also has the universal property of the amalgam RHS and hence there must be a unique isomorphism between the two. Consider the diagram below:

With the amalgam ${G_1 *_{A_1} G_2}$ we have homomorphisms ${\eta_1: G_1 \rightarrow G_1 *_{A_1} G_2}$ and ${\eta_2 : G_2 \rightarrow G_2 *_{A_1} G_2}$ and the with amalgam ${(G_1 *_{A_1} G_2) *_{A_2} G_3}$ we have homomorphisms ${\xi_1: G_1 *_{A_1} G_2 \rightarrow (G_1 *_{A_1} G_2) *_{A_2} G_3}$ .

Now if ${H}$ is any group with maps ${h_i: G_i\rightarrow H}$ for ${i=1,2,3}$ , that are compatible with ${f_i:A_i \rightarrow G_i}$ , then we need to show that ${h_i}$ factors via a unique homomorphism ${\Phi:(G_1 *_{A_1} G_2) *_{A_2} G_3 \rightarrow H}$ . ${h_i}$ .

The map ${h_1 : G_1 \rightarrow H}$ and ${h_2 : G_2 \rightarrow H}$ are compatible with ${f_1}$ and ${f_1'}$ so by universal property of the amalgam ${G_1 *_{A_1} G_2}$ we have a unique morphism ${\varphi : G_1 *_{A_1} G_2 \rightarrow H}$ such that ${h_1 =\varphi \circ i_1}$ . Now since ${\varphi, h_3}$ are compatible with ${\eta_2 \circ f_2}$ and ${f_2'}$ , the map ${\varphi}$ can be further factored using the universal property of the amalgam ${(G_1 *_{A_1} G_2) *_{A_2} G_3}$ by the unique homomorphism ${\Phi: (G_1 *_{A_1} G_2) *_{A_2} G_3 \rightarrow H}$ . This shows the associativity of amalgamation.

Note 1: When amalgams were first introduced, before category theory came into being, (see for example ww-2 era papers by B.H Neumann on free groups with amalgamation et. al), they were treated as generalized versions of free products of groups with complicated inner structures. They never made use of these elegant universal properties and proving such things as associativity, can get really messy.

Updated:Fixed typo

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