[[Group theory MOC]]
# Free product of groups

The **free product** is the [[Products and coproducts|coproduct]] in [[Category of groups]]. #m/def/group
Let $\{ G_{i} \}_{i \in G}$ be groups.
Then the free product has finite strings of elements taken from any $G_{i}$ as its [[Word (algebra)|words]] with all the identities considered the same, 
which may then be reduced using the group laws of each $G_{i}$.
A reduced word is therefore a word where no adjacent elements come from the same group.
Reduced words may be regarded as the elements of the free product group

A more general construction is the [[Amalgamated free product]].

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