Let be a group, and let be a family of subgroups (http://planetmath.org/Subgroup) of . Then is said to be a free product of the subgroups if given any group and a homomorphism (http://planetmath.org/GroupHomomorphism) for each , there is a unique homomorphism such that for all . The subgroups are then called the free factors of .
If is the free product of , and is a family of groups such that for each , then we may also say that is the free product of . With this definition, every family of groups has a free product, and the free product is unique up to isomorphism.
The free product is the coproduct in the category of groups.
Free groups 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 (http://planetmath.org/QuotientGroup) of a free group.
Let be a family of groups. For each , let be a set and a function such that generates . The should be chosen to be pairwise disjoint; for example, we could take , and let be the obvious bijection. Let be a free group freely generated by . For each , the subgroup of is freely generated by , so there is a homomorphism extending . Let be the normal closure of in .
Then it can be shown that is the free product of the family of subgroups , and for each .
|Date of creation||2013-03-22 14:53:34|
|Last modified on||2013-03-22 14:53:34|
|Last modified by||yark (2760)|