| Version 2 |
Version 1 |
| \PMlinkescapeword{homomorphism} |
Let $G$ be a group, and let $(A_i)_{i\in I}$ be a family of subgroups of $G$. |
| \PMlinkescapeword{subgroup} |
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| Let $G$ be a group, and let $(A_i)_{i\in I}$ be a family of \Pmlinkname{subgroups}{Subgroup} of $G$. |
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| Then $G$ is said to be a \emph{free product} of the subgroups $A_i$ |
Then $G$ is said to be a \emph{free product} of the subgroups $A_i$ |
| if given any group $H$ and |
if given any group $H$ and |
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a \PMlinkname{homomorphism}{GroupHomomorphism} $f_i\colon A_i\to H$ for each $i\in I$,
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a homomorphism $f_i\colon A_i\to H$ for each $i\in I$,
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| there is a unique homomorphism $f\colon G\to H$ |
there is a unique homomorphism $f\colon G\to H$ |
| such that $f|_{A_i}=f_i$ for all $i\in I$. |
such that $f|_{A_i}=f_i$ for all $i\in I$. |
| The subgroups $A_i$ are then called the \emph{free factors} of $G$. |
The subgroups $A_i$ are then called the \emph{free factors} of $G$. |
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| If $G$ is the free product of $(A_i)_{i\in I}$, |
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 $K_i\isomorphic A_i$ |
and $(K_i)_{i\in I}$ is a family of groups such that $K_i\isomorphic A_i$ |
| for each $i\in I$, |
for each $i\in I$, |
| then we may also say that $G$ is the free product of $(K_i)_{i\in I}$. |
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, |
With this definition, every family of groups has a free product, |
| and the free product is unique up to isomorphism. |
and the free product is unique up to isomorphism. |
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| Free groups are simply the free products of infinite cyclic groups. |
Free groups are simply the free products of infinite cyclic groups. |
