nLab free product of groups

Idea

Given a pair of groups $G_1, G_2 \,\in\,$ Grp, or more generally an $n$-tuple $\big(G_1, G_2, \cdots, G_n\big)$ of groups, what is traditionally called their free product and denoted

$$G_1 \star G_2 \;\;\; \in \; Grp$$

or generally

$$\star_i G_i \;\coloneqq\; G_1 \star G_2 \star \cdots \star G_n \;\;\; \in \; Grp$$

is really their coproduct in Grp:

$$\star_i G_i \;\simeq\; \amalg_i G_i \;\;\; \in \; Grp \,.$$

(Compare the dual notion of “direct product of groups”, which are really category theoretic products of groups.)

Informally, the “free product” is the group whose elements are freely generated form those of the $G_i$, subject only to the relations given by the group operations in each of these groups.

More generally, if each of the given groups is equipped with a homomorphism

$$\iota_i \,\colon\, H \longrightarrow G_i$$

from a fixed group $H \,\in\, Grp$

(often taken to be monomorphisms, hence injections, hence subgroup-inclusions $A \hookrightarrow G_i$ and reducing to the previous situation when $H = 1$ is the trivial group)

then what is traditionally called their amalgamated free product (or similar), and denoted

$$G_1 \star_A G_2 \;\;\; \in \; Grp$$

or generally

$$\underset{A}{\star_i} G_i \;\coloneqq\; G_1 \star_A G_2 \star_A \cdots \star_A G_n \;\;\; \in \; Grp$$

is the corresponding pushout

$$G_1 \star_A G_2 \;\; \simeq \;\; G_1 \overset{H}{\amalg} G_2 \;\; \in \; Grp$$

(or generally the colimit) in the category Grp, hence the unique group, up to isomorphism, which

1. receives homomorphisms $q_i$ from the $G_i$ such that their precompositions with the respective $\iota_i$ all agree,

2. is universal with this property in that for any other group $Q$ receiving such homomorphisms $f_i$ these factor through the respective $q_i$ via a single and unique comparison homomorphism, shown as a dashed arrow in the following diagram:

Examples

Example

For $S \in$ Sets, the free group on $S$ is equivalently the free product of $S$ copies of the additive group of integers:

$$F(S) \;\simeq\; \star_S \mathbb{Z} \,.$$

An abstract way to understand this elementary fact is to notice that $F$ is the free construction left adjoint to the forgetful functor which sends a group $G$ to its underlying set $U(G)$:

$$F \,\colon\,Set \rightleftarrows Grp \,\colon\, U \,.$$

But since

1. left adjoints preserve colimits and hence in particular preserve coproducts,

2. every set is the coproduct of copies of the singleton set $\ast \,\in\,$ Set indexed by its elements $s \in S$,

3. the free group on the singleton set is the additive group of integers

we have

$$\begin{array}{l} F(S) \\ \;\simeq\; F\Big(\underset{s \in S}{\sqcup} \ast\Big) \\ \;\simeq\; \underset{s \in S}{\coprod} F(\ast) \\ \;\simeq\; \underset{s \in S}{\coprod} \mathbb{Z} \\ \;\simeq\; \underset{s \in S}{\star} \mathbb{Z} \,. \end{array}$$

Properties

Presentation

If all the $G_i$ have group presentations, then their free product $\star_i G_i$ has a group presentation as follows:

Assume $G_i = \langle S_i|R_i\rangle = F_i/N_i$, where

1. each $F_i=\langle S_i\rangle$ is the free group on a set $S_i$

2. $N_i\subset F_i$ is the normal subgroup of $F_i$ generated by the subset $R_i\subset F_i$,

then the free product

$$\star_i G_i \;\coloneqq\; \left\langle \coprod_i S_i | \coprod_i R_i \right\rangle \;=\; (\star_i F_i)/\langle\cup_i N_i\rangle$$

is presented by the disjoint unions of the $S_i$ and the $R_i$.

(As with anything satisfying a universal property, the result does not depend on the presentations, up to unique isomorphism.)

Existence

Free products of groups always exist.

With the above discussion of presentations this follows from the fact that any group $G_i$ has a presentation: we can always take $S_i$ to be the underlying set of $G_i$ and take $R_i$ to be the set of all words in $F_i$ that become equal to the neutral element in $G_i$.

The value of the more general construction above is that one often has much smaller $S_i$ and $R_i$ to work with. Even if $G_i$ is infinite, the $S_i$ and $R_i$ might be finite (in the strictest sense), making this part of finite mathematics and directly subject to the methods of combinatorial group theory.

Related concepts

• free group

• pushout

• amalgamation

• direct product of groups

• semidirect product group

• central product of groups

References

The notion of amalgamated free products of groups seems to be due to

• Otto Schreier, §1 in: Die Untergruppen der freien Gruppen, Abh. Math. Semin. Univ. Hambg. 5 (1927) 161–183 [doi:10.1007/BF02952517]

where it is discussed (though not under any name) complete with the universal property of what today we recognize as a colimit (long before this category theory-notion was formulated in generality).

The actual terminology “amalgamated free product” seems to be due to:

• Hanna Neumann, Generalized Free Products with Amalgamated Subgroups, American Journal of Mathematics, 71 3 (1949) 491-540 [jstor:2372346]

• B. Hanna Neumann, An Essay on Free Products of Groups with Amalgamations, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 246 919 (1954) 503-554 [jstor:91573]

Textbook accounts of free products of groups:

• Anthony Knapp, Section VII.3 of: Basic Algebra, Springer (2006) [doi:10.1007/978-0-8176-4529-8, pdf]

Expository review in the generality of amalgamated free products and making explicit their nature as pushouts in Grp:

• Abhay Chandel, Free groups and amalgamated product, BSc thesis (2013) [pdf, pdf]

See also

• eom, Amalgam of groups

Discussion of subgroups of amalgamated from products:

• A. Karrass, D. Solitar, The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup, Transactions of the American Mathematical Society 150 1 (1970) 227-255 [doi:10.2307/1995492]