What does it mean to say a single element acts semi-regularly

Question

Let $G$ be a group acting on some set $\Omega$. Just a minor point, but saying that some nontrivial element $g \in G$ acts semi-regularly, does this mean that $g$ itself has no fixed point, or that the subgroup generated by $g$ has trivial point stabilizers for each point? I guess the latter, but just to confirm my guess I am asking.

Answer 1

The usual definition I've seen is that an element is said to be semiregular (or act semiregularly) if it generates a semiregular subgroup. It is not hard to see that this is equivalent to the element having all cycles of the same length, when written as a product of disjoint cycles.

Answer 2

A group $G$ acts semiregularly on a set $X$, if it acts freely, i.e., for $g\in G$, if there exists an $x$ in $X$ with $g.x = x$ (that is, if $g$ has at least one fixed point), then $g$ is the identity. There are several types of group actions, e.g., see here.