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Construction

If G and H are groups, a word on G and H is a sequence of the form

s 1 s 2 ⋯ s n , {\displaystyle s_{1}s_{2}\cdots s_{n},}

where each si is either an element of G or an element of H. Such a word may be reduced using the following operations:

• Remove an instance of the identity element (of either G or H).
• Replace a pair of the form g1g2 by its product in G, or a pair h1h2 by its product in H.

Every reduced word is either the empty sequence, contains exactly one element of G or H, or is an alternating sequence of elements of G and elements of H, e.g.

g 1 h 1 g 2 h 2 ⋯ g k h k . {\displaystyle g_{1}h_{1}g_{2}h_{2}\cdots g_{k}h_{k}.}

The free product G ∗ H is the group whose elements are the reduced words in G and H, under the operation of concatenation followed by reduction.

For example, if G is the infinite cyclic group ⟨ x ⟩ {\displaystyle \langle x\rangle } , and H is the infinite cyclic group ⟨ y ⟩ {\displaystyle \langle y\rangle } , then every element of G ∗ H is an alternating product of powers of x with powers of y. In this case, G ∗ H is isomorphic to the free group generated by x and y.

Presentation

Suppose that

G = ⟨ S G ∣ R G ⟩ {\displaystyle G=\langle S_{G}\mid R_{G}\rangle }

is a presentation for G (where SG is a set of generators and RG is a set of relations), and suppose that

H = ⟨ S H ∣ R H ⟩ {\displaystyle H=\langle S_{H}\mid R_{H}\rangle }

is a presentation for H. Then

G ∗ H = ⟨ S G ∪ S H ∣ R G ∪ R H ⟩ . {\displaystyle G*H=\langle S_{G}\cup S_{H}\mid R_{G}\cup R_{H}\rangle .}

That is, G ∗ H is generated by the generators for G together with the generators for H, with relations consisting of the relations from G together with the relations from H (assume here no notational clashes so that these are in fact disjoint unions).

Examples

For example, suppose that G is a cyclic group of order 4,

G = ⟨ x ∣ x 4 = 1 ⟩ , {\displaystyle G=\langle x\mid x^{4}=1\rangle ,}

and H is a cyclic group of order 5

H = ⟨ y ∣ y 5 = 1 ⟩ . {\displaystyle H=\langle y\mid y^{5}=1\rangle .}

Then G ∗ H is the infinite group

G ∗ H = ⟨ x , y ∣ x 4 = y 5 = 1 ⟩ . {\displaystyle G*H=\langle x,y\mid x^{4}=y^{5}=1\rangle .}

Because there are no relations in a free group, the free product of free groups is always a free group. In particular,

F m ∗ F n ≅ F m + n , {\displaystyle F_{m}*F_{n}\cong F_{m+n},}

where Fn denotes the free group on n generators.

Another example is the modular group P S L 2 ( Z ) {\displaystyle PSL_{2}(\mathbf {Z} )} . It is isomorphic to the free product of two cyclic groups:¹

P S L 2 ( Z ) ≅ ( Z / 2 Z ) ∗ ( Z / 3 Z ) . {\displaystyle PSL_{2}(\mathbf {Z} )\cong (\mathbf {Z} /2\mathbf {Z} )\ast (\mathbf {Z} /3\mathbf {Z} ).}

Generalization: Free product with amalgamation

The more general construction of free product with amalgamation is correspondingly a special kind of pushout in the same category. Suppose G {\displaystyle G} and H {\displaystyle H} are given as before, along with monomorphisms (i.e. injective group homomorphisms):

φ : F → G   {\displaystyle \varphi :F\rightarrow G\ \,} and   ψ : F → H , {\displaystyle \ \,\psi :F\rightarrow H,}

where F {\displaystyle F} is some arbitrary group. Start with the free product G ∗ H {\displaystyle G*H} and adjoin as relations

φ ( f ) ψ ( f ) − 1 = 1 {\displaystyle \varphi (f)\psi (f)^{-1}=1}

for every f {\displaystyle f} in F {\displaystyle F} . In other words, take the smallest normal subgroup N {\displaystyle N} of G ∗ H {\displaystyle G*H} containing all elements on the left-hand side of the above equation, which are tacitly being considered in G ∗ H {\displaystyle G*H} by means of the inclusions of G {\displaystyle G} and H {\displaystyle H} in their free product. The free product with amalgamation of G {\displaystyle G} and H {\displaystyle H} , with respect to φ {\displaystyle \varphi } and ψ {\displaystyle \psi } , is the quotient group

( G ∗ H ) / N . {\displaystyle (G*H)/N.\,}

The amalgamation has forced an identification between φ ( F ) {\displaystyle \varphi (F)} in G {\displaystyle G} with ψ ( F ) {\displaystyle \psi (F)} in H {\displaystyle H} , element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a path-connected subspace, with F {\displaystyle F} taking the role of the fundamental group of the subspace. See: Seifert–van Kampen theorem.

Karrass and Solitar have given a description of the subgroups of a free product with amalgamation.² For example, the homomorphisms from G {\displaystyle G} and H {\displaystyle H} to the quotient group ( G ∗ H ) / N {\displaystyle (G*H)/N} that are induced by φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are both injective, as is the induced homomorphism from F {\displaystyle F} .

Free products with amalgamation and a closely related notion of HNN extension are basic building blocks in Bass–Serre theory of groups acting on trees.

In other branches

One may similarly define free products of other algebraic structures than groups, including algebras over a field. Free products of algebras of random variables play the same role in defining "freeness" in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory.

See also

• Direct product of groups
• Coproduct
• Graph of groups
• Kurosh subgroup theorem
• Normal form for free groups and free product of groups
• Universal property

References

1. Alperin, Roger C. (April 1993). "PSL2(Z) = Z2 * Z3". Amer. Math. Monthly. 100: 385–386. doi:10.1080/00029890.1993.11990418. /wiki/Roger_C._Alperin ↩

2. A. Karrass and D. Solitar (1970) The subgroups of a free product of two groups with an amalgamated subgroup, Transactions of the American Mathematical Society 150: 227–255. https://www.ams.org/journals/tran/1970-150-01/S0002-9947-1970-0260879-9/S0002-9947-1970-0260879-9.pdf ↩