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'free product'
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| Title of object: |
free product |
| Canonical Name: |
FreeProduct |
| Type: |
Definition |
| Created on: |
2004-12-13 15:25:38 |
| Modified on: |
2004-12-13 15:29:51 |
| Classification: |
msc:20E06 |
| Defines: |
free factor |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\def\isomorphic{\cong} |
Content:
\PMlinkescapeword{homomorphism}
\PMlinkescapeword{subgroup}
Let $G$ be a group, and let $(A_i)_{i\in I}$ be a family of \Pmlinkname{subgroups}{Subgroup} of $G$.
Then $G$ is said to be a \emph{free product} of the subgroups $A_i$
if given any group $H$ and
a \PMlinkname{homomorphism}{GroupHomomorphism} $f_i\colon A_i\to H$ for each $i\in I$,
there is a unique homomorphism $f\colon G\to H$
such that $f|_{A_i}=f_i$ for all $i\in I$.
The subgroups $A_i$ are then called the \emph{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 $K_i\isomorphic A_i$
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.
Free groups are simply the free products of infinite cyclic groups. |
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