Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.
2026-03-25 19:03:43.1774465423
Are free groups and free actions related?
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Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $I\to G$ meeting once each orbit; the obvious way to produce it is just considering $X=G\times I$ with action $g\cdot (h,i)=(gh,i)$ and the map $I\to X$, $i\mapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.