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free product (Definition)

Definition

Let $ G$ be a group, and let $ (A_i)_{i\in I}$ be a family of subgroups of $ G$. Then $ G$ is said to be a free product of the subgroups $ A_i$ if given any group $ H$ and a homomorphism $ f_i\colon A_i\to H$ for each $ i\in I$, there is a unique homomorphism $ f\colon G\to H$ such that $ f\vert _{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 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 $ (K_i)_{i\in I}$ be a family of groups. For each $ i\in I$, let $ X_i$ be a set and $ \gamma_i\colon X_i\to K_i$ a function such 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 $ {\langle X_i\rangle}$ of $ F$ is freely generated by $ X_i$, so there is a homomorphism $ \phi_i\colon{\langle X_i\rangle}\to K_i$ extending $ \gamma_i$. Let $ N$ be the normal closure of $ \bigcup_{i\in I}\ker{\phi_i}$ in $ F$.

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



"free product" is owned by yark.
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See Also: free product with amalgamated subgroup, free group

Also defines:  free factor

Attachments:
free products and group actions (Theorem) by rm50
$PSL_2(\mathbb{Z})$ is a free product (Example) by rm50
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Cross-references: normal closure, freely generated, bijection, pairwise disjoint, function, infinite cyclic groups, free groups, category, coproduct, isomorphism, group
There are 8 references to this entry.

This is version 9 of free product, born on 2004-12-13, modified 2004-12-14.
Object id is 6574, canonical name is FreeProduct.
Accessed 3933 times total.

Classification:
AMS MSC: 20E06 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations)
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