# Why is a free group action called free?

A group action is called free if every element other than the identity in the group has no fixed points. What does "free" mean? Not able to connect the name to the math definition, I am afraid I am missing something.

• ''Free'' is used in the sense that the group elements do not have any nontrivial interaction. Dec 31, 2021 at 12:16
• I do not know whether this is historically accurate, but I'd assume the name "free" just comes as short form of "fixed-point free", i.e. "without fixed points", in this context. Dec 31, 2021 at 12:42
• A free action of $G$ on $X$ essentially means that $X$ can be identified with a disjoint union of copies of $G$ where $G$ acts on each copy of itself by left-multiplication. Every (other) $G$-set can be viewed as a quotient (orbit-wise) of such a free $G$-set. Dec 31, 2021 at 13:27
• "What does "free" mean"? This question can be answered by yourself. Look up the definition of, say, a free group or a free algebra. Dec 31, 2021 at 13:46