The free product is the coproduct in Category of groups. group
Let {πΊπ}πβπΊ be groups.
Then the free product has finite strings of elements taken from any πΊπ as its words with all the identities considered the same,
which may then be reduced using the group laws of each πΊπ.
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