Free product of groups

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A group generated by the groups such that any homomorphisms of the into an arbitrary group can be extended to a homomorphism . The symbol * is used to denote a free product, for example,

in the case of a finite set . Each element of a free product that is not the identity can be expressed uniquely as an irreducible word , where , and for any . The construction of a free product is important in the study of groups defined by a set of generating elements and defining relations. In these terms it can be defined as follows. Suppose that each group is defined by sets of generators and of defining relations, where if . Then the group defined by the set of generators and the set of defining relations is the free product of the groups , .

Every subgroup of a free product can be decomposed into a free product of subgroups, of which some are infinite cyclic and each of the others is conjugate with some subgroup of a group in the free decomposition of (Kurosh' theorem).


[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)


The notion of a free product of groups is a special case of that of a free product with amalgamated subgroup (see Amalgam of groups).

How to Cite This Entry:
Free product of groups. A.L. Shmel'kin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098