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	Difficulty level	Fact about
Abelian implies every subgroup is potentially characteristic		Abelian group (?) Subgroup (?) Potentially characteristic subgroup (?) Subgroup of abelian group (2) Potentially characteristic subgroup of abelian group (3)
Commutator of a group and a subgroup implies normal	3	Commutator operator (3) Improper subgroup (2) Subgroup (2) Normal subgroup (2) Subgroup realizable as the commutator of the whole group and a subgroup (2)
Equivalence of definitions of subgroup		Subgroup (1)
Every subgroup has a left transversal		Subgroup (?) Left coset of a subgroup (?)
Every subgroup is contracharacteristic in its normal closure		Composition operator (?) Contracharacteristic subgroup (?) Normal subgroup (?) Subgroup (?)
Lagrange's theorem	2	Order of a group (1) Subgroup (1) Group (1) Size (1) Left coset of a subgroup (1)
Quotient group need not be isomorphic to any subgroup		Normal subgroup (2) Endomorphism kernel (2) Subgroup (2) Quotient map (2)
Subgroup need not be isomorphic to any quotient group		Subgroup (?) Quotient map (?)
Subset containing identity whose left translates partition the group is a subgroup		Subgroup (1)
Sufficiency of subgroup criterion		Subgroup (1)

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