,"Quick phrase","Defining ingredient"
"2-Engel group",,"Normal closure,Normal subgroup generated by a subset,Abelian group,Abelian normal subgroup,Local nilpotency class,Group of nilpotency class two,Bounded Engel group,Levi operator"
"Abelian Sylow subgroup",,"Sylow subgroup,Abelian group"
"Abelian characteristic subgroup",,"Characteristic subgroup,Abelian group"
"Abelian conjugacy-closed subgroup",,"Conjugacy-closed subgroup,Abelian group"
"Abelian critical subgroup",,"Critical subgroup,Abelian group"
"Abelian direct factor",,"Direct factor,Abelian group,Central subgroup"
"Abelian fully invariant subgroup",,"Fully invariant subgroup,Abelian group"
"Abelian hereditarily normal subgroup",,"Hereditarily normal subgroup,Abelian group,Transitively normal subgroup,Abelian normal subgroup"
"Abelian marginal subgroup",,"Marginal subgroup,Abelian group"
"Abelian normal subgroup",,"Normal subgroup,Abelian group"
"Abelian normal subgroup of maximum rank",,"Abelian normal subgroup,Abelian group,Normal subgroup,Rank of a p-group,Normal rank of a p-group"
"Abelian pronormal subgroup",,"Pronormal subgroup,Abelian group"
"Abelian subgroup",,"Abelian group"
"Abelian subnormal subgroup",,"Subnormal subgroup,Abelian group"
"Abelian verbal subgroup",,"Verbal subgroup,Abelian group"
"Associative ring",,"Abelian group"
CA-group,,"Centralizer,Abelian group,Centerless group,Center"
"Central subgroup",,"Central factor,Abelian group"
"Central subgroup of normalizer",,"Central factor of normalizer,Abelian group,Central subgroup,Center,In-normalizer operator"
"Characteristic direct factor of abelian group",,"Characteristic direct factor,Abelian group,Fully invariant direct factor"
"Characteristic subgroup of abelian group",,"Characteristic subgroup,Abelian group"
"Constant APS of an abelian group",,"Abelian group"
"Divisibility-closed subgroup of abelian group",,"Divisibility-closed subgroup,Abelian group"
"Division ring",,"Abelian group,Associative binary operation"
Field,,"Abelian group"
"Finite group that is order statistics-equivalent to an abelian group",,"Order statistics-equivalent finite groups,Order statistics of a finite group,Abelian group,Finite abelian group"
"Fully invariant subgroup of abelian group",,"Fully invariant subgroup,Abelian group"
"Group admitting a partition into abelian subgroups",,"Abelian subgroup,Abelian group"
"Group having an abelian contranormal subgroup",,"Abelian group,Contranormal subgroup,Normal closure"
"Group of nilpotency class two","class two,inner automorphism group is abelian,commutator subgroup inside center,derived subgroup inside center,commutators are central,triple commutators are trivial","Nilpotency class,Derived subgroup,Center,Inner automorphism group,Abelian group"
"Group that is 1-isomorphic to an abelian group",,"1-isomorphic groups,Abelian group"
"Group whose automorphism group is abelian","abelian automorphism group,any two automorphisms commute","Automorphism group of a group,Abelian group,Normal subgroup,Normal subgroup contained in centralizer of derived subgroup"
"Group whose center is a direct factor",,"Centerless group,Abelian group,Center,Permutably complemented subgroup,Complemented normal subgroup,Direct factor"
"Lie ring of nilpotency class two","class two,inner derivation Lie ring is abelian,derived subring inside center,Lie brackets are central,triple Lie brackets are trivial","Nilpotency class,Derived subring of a Lie ring,Center,Inner derivation,Abelian group"
"Maximal among abelian subgroups",,"Self-centralizing subgroup,Abelian group,Maximal operator,Abelian subgroup"
"Metabelian group","abelian-by-abelian,abelian normal subgroup with abelian quotient,abelian derived subgroup,abelian commutator subgroup","Solvable group,Derived length,Derived subgroup,Abelian group,Abelian normal subgroup,Abelian-quotient subgroup,Commutator,Meta operator"
"Non-associative ring",,"Abelian group"
"Powering-invariant subgroup of abelian group",,"Powering-invariant subgroup,Abelian group"
"Residually finite abelian group",,"Residually finite group,Abelian group,Conjugacy-separable group"
"Solvable group",,"Normal series,Abelian group,Subnormal series,Derived series,Derived subgroup,Characteristic series,Fully invariant series"
"Subgroup of abelian group",,"Subgroup,Abelian group"
"Verbal subgroup of abelian group",,"Verbal subgroup,Abelian group"
"Virtually abelian group",,"Virtually operator,Abelian group"