query: In the context of free or faithful group actions, what is the stabilizer subgroup when the set is empty?
positive_passages.docid: 66464
positive_passages.text: Faithful group actions for singleton group or for empty set
negative_passages.docid: 144788
negative_passages.text: Cosets of a subgroup do not overlap
negative_passages.docid: 99346
negative_passages.text: Prove a result on transitive group actions.
negative_passages.docid: 39895
negative_passages.text: Showing a free abelian group is generated by its basis
negative_passages.docid: 2761
negative_passages.text: If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
negative_passages.docid: 27436
negative_passages.text: Does $G\cong G/H$ imply that $H$ is trivial?
negative_passages.docid: 6466
negative_passages.text: Image of subgroup and Kernel of homomorphism form subgroups
negative_passages.docid: 54346
negative_passages.text: If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open?
negative_passages.docid: 71653
negative_passages.text: Why is the Cartesian product of a set $A$ and empty set an empty set?
negative_passages.docid: 58076
negative_passages.text: Subgroup generated by a set
negative_passages.docid: 10602
negative_passages.text: Monotone Union of subgroups being subgroup
negative_passages.docid: 117479
negative_passages.text: Mutual set of representatives for left and right cosets: what about infinite groups?
negative_passages.docid: 18181
negative_passages.text: Complete classification of the groups for which converse of Lagrange's Theorem holds
negative_passages.docid: 93666
negative_passages.text: Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
negative_passages.docid: 70298
negative_passages.text: Are quotients of chain groups $C_n(X)/C_n(A)$ still free?
negative_passages.docid: 117701
negative_passages.text: Let $G$ be a group and $X$ set. $a,b \in X$ are in the same orbit, so show that $stab(a) \cong stab (b)$
negative_passages.docid: 98474
negative_passages.text: Show the existence of a certain subgroup of F2