free product

Definition

Let $G$ be a group, and let ${(A_i)}_{i\in I}$ bea family of subgroups (http://planetmath.org/Subgroup) of $G$.Then $G$ is said to be a free product of the subgroups $A_i$if given any group $H$ anda homomorphism (http://planetmath.org/GroupHomomorphism)$f_i:A_i\to H$ for each $i\in I$,there is a unique homomorphism $f:G\to H$such that ${f|}_{A_i}=f_i$ for all $i\in I$.The subgroups $A_i$ are then called the free factors of $G$.

If $G$ is the free product of ${(A_i)}_{i\in I}$,and ${(K_i)}_{i\in I}$ is a family of groups such that $K_i\cong A_i$for each $i\in I$,then we may also say that $G$ is the free product of ${(K_i)}_{i\in I}$.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.

Construction

Free groups are simply the free products of infinite cyclic groups,and it is possible to generalize the construction given in the free grouparticle to the case of arbitrary free products.But we will instead constructthe free product as a quotient (http://planetmath.org/QuotientGroup) of a free group.

Let ${(K_i)}_{i\in I}$ be a family of groups.For each $i\in I$,let $X_i$ be a set and ${\gamma }_i:X_i\to K_i$ a functionsuch that ${\gamma }_i(X_i)$ generates $K_i$.The $X_i$ should be chosen to be pairwise disjoint;for example, we could take $X_i=K_i\times \{i\}$,and let ${\gamma }_i$ be the obvious bijection.Let $F$ be a free group freely generated by ${\bigcup }_{i\in I}{X}_i$.For each $i\in I$,the subgroup $⟨X_i⟩$ of $F$ is freely generated by $X_i$,so there is a homomorphism ${\varphi }_i:⟨X_i⟩\to K_i$extending ${\gamma }_i$.Let $N$ be the normal closure of ${\bigcup }_{i\in I}\ker {\varphi }_i$ in $F$.

Then it can be shown that $F/N$ isthe free product of the family of subgroups ${(⟨X_i⟩N/N)}_{i\in I}$,and $K_i\cong ⟨X_i⟩N/N$ for each $i\in I$.