nLab free product of groups



Group Theory

Limits and colimits



Given a pair of groups G 1,G 2G_1, G_2 \,\in\, Grp, or more generally an n n -tuple (G 1,G 2,,G n)\big(G_1, G_2, \cdots, G_n\big) of groups, what is traditionally called their free product and denoted

G 1G 2Grp G_1 \star G_2 \;\;\; \in \; Grp

or generally

iG iG 1G 2G nGrp \star_i G_i \;\coloneqq\; G_1 \star G_2 \star \cdots \star G_n \;\;\; \in \; Grp

is really their coproduct in Grp:

iG i⨿ iG iGrp. \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 iG_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

ι i:HG i \iota_i \,\colon\, H \longrightarrow G_i

from a fixed group HGrpH \,\in\, Grp

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

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

G 1 AG 2Grp G_1 \star_A G_2 \;\;\; \in \; Grp

or generally

iAG iG 1 AG 2 A AG nGrp \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 AG 2G 1⨿HG 2Grp 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 iq_i from the G iG_i such that their precompositions with the respective ι i\iota_i all agree,

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



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

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

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

F:SetGrp:U. 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 sSs \in S,

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

we have

F(S) F(sS*) sSF(*) sS sS. \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}



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

Assume G i=S i|R i=F i/N iG_i = \langle S_i|R_i\rangle = F_i/N_i, where

  1. each F i=S iF_i=\langle S_i\rangle is the free group on a set S iS_i

  2. N iF iN_i\subset F_i is the normal subgroup of F iF_i generated by the subset R iF iR_i\subset F_i,

then the free product

iG i iS i| iR i=( iF i)/ iN i \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 iS_i and the R iR_i.

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


Free products of groups always exist.

With the above discussion of presentations this follows from the fact that any group G iG_i has a presentation: we can always take S iS_i to be the underlying set of G iG_i and take R iR_i to be the set of all words in F iF_i that become equal to the neutral element in G iG_i.

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


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

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:

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

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]

Last revised on May 19, 2023 at 05:54:00. See the history of this page for a list of all contributions to it.