It would be awesome to have the details written out of the case of the symmetric group on the empty set.
Searching Google Books (and Wikipedia), I don't see precedents for use of "freely reduced" rather than "reduced".
Now, inventing and using good terminology is a good thing. What do you think about expanding on the choice of term in the article text?
It would be awesome to have the details written out of the case of the symmetric group on the empty set.