A group action is regular if and only if free and transitive. I have questions on the non-empty part.
66464
Faithful group actions for singleton group or for empty set
43711
Categorical Quotient and group actions
66510
A group is abelian if and only if the center of the group is all the group
39100
Exercise from Serre's "Trees" - prove that a given group is trivial
138436
How to prove reflexivity, symmetry and transitivity for the following relation?
33868
A transformation between two free group actions.
123282
free product with amalgamation is correspondingly a pushout
22629
rank of the free group: $\mathrm{rank} F_X=|X|$
111348
Show that $\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$
77297
Counterexample for $A,B\triangleleft G,G/A\cong B \Rightarrow G/B\cong A$
14301
Union of subgroups is a subgroup if and only if one subgroup is a subset of the other
8066
Sort-of-simple non-Hopfian groups
128711
$(\Bbb Z\oplus \Bbb Z)[X] \not\cong \Bbb Z[X]\oplus \Bbb Z[X]\,$?
132704
If G is not commutative, then is there always a subgroup that is not a normal subgroup?
133762
A set which satisfies all conditions for a Group except associativity
156910
Necessary and sufficient condition for $R/IJ \cong R/I \times R/J$
24561
$\mathbb S_n$ as semidirect product