free product


Let G be a group, and let (Ai)iI be a family of subgroupsMathworldPlanetmathPlanetmath ( of G. Then G is said to be a free productMathworldPlanetmath of the subgroups Ai if given any group H and a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( fi:AiH for each iI, there is a unique homomorphism f:GH such that f|Ai=fi for all iI. The subgroups Ai are then called the free factors of G.

If G is the free product of (Ai)iI, and (Ki)iI is a family of groups such that KiAi for each iI, then we may also say that G is the free product of (Ki)iI. With this definition, every family of groups has a free product, and the free product is unique up to isomorphismMathworldPlanetmathPlanetmath.

The free product is the coproductMathworldPlanetmath in the category of groups.


Free groupsMathworldPlanetmath are simply the free products of infinite cyclic groups, and it is possible to generalize the construction given in the free group article to the case of arbitrary free products. But we will instead construct the free product as a quotient ( of a free group.

Let (Ki)iI be a family of groups. For each iI, let Xi be a set and γi:XiKi a function such that γi(Xi) generates Ki. The Xi should be chosen to be pairwise disjoint; for example, we could take Xi=Ki×{i}, and let γi be the obvious bijection. Let F be a free group freely generated by iIXi. For each iI, the subgroup Xi of F is freely generated by Xi, so there is a homomorphism ϕi:XiKi extending γi. Let N be the normal closurePlanetmathPlanetmath of iIkerϕi in F.

Then it can be shown that F/N is the free product of the family of subgroups (XiN/N)iI, and KiXiN/N for each iI.

Title free product
Canonical name FreeProduct
Date of creation 2013-03-22 14:53:34
Last modified on 2013-03-22 14:53:34
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 20E06
Related topic FreeProductWithAmalgamatedSubgroup
Related topic FreeGroup
Defines free factor