Assertion: abelian group's subgroup | /r/HasProperty | trivial
Facets:
Source sentences:
text: Simple group is that it is not trivial and subgroups of simple group must be improper and trivial.
text: A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.[s]
text: A torsion-free abelian group has no non-trivial finite subgroups.
text: Every finite group whose order is divisible by p contains a subgroup which is a non-trivial p-group, namely a cyclic group of order p generated by an element of order p obtained from Cauchy's theorem.
text: Since any subgroup-defining function (such as the center, the derived subgroup, the Frattini subgroup, the Fitting subgroup etc.) returns a characteristic subgroup of the whole group, and since every characteristic subgroup is normal, any subgroup obtained via a subgroup-defining function must be either trivial or the whole group.
text: So, every simple abelian group has trivial subgroup and improper subgroup.