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	Quick phrase	Defining ingredient
Cyclic characteristic subgroup		Characteristic subgroup Cyclic group
Cyclic group:Z36		Cyclic group Finite cyclic group External direct product Cyclic group:Z4
Cyclic isomorph-containing subgroup		Isomorph-containing subgroup Cyclic group
Cyclic normal subgroup		Normal subgroup Cyclic group
Cyclicity-forcing number		Order of a group Cyclic group Euler totient function Square-free number Abelianness-forcing number Nilpotency-forcing number
Dedekind group	every subgroup is normal every cyclic subgroup is normal normal closure of element is cyclic	Normal subgroup Inner automorphism Power automorphism Normal closure Levi operator Cyclic group Hamiltonian operator
Generalized symmetric group		External wreath product Cyclic group Symmetric group Symmetric group on finite set External semidirect product Homocyclic group
Group in which every finite subgroup is cyclic		Finite subgroup Cyclic group Finite cyclic subgroup
Group whose outer automorphism group is cyclic		Outer automorphism group Cyclic group Inner automorphism group 1-completed subgroup Automorphism group of a group
Metacyclic group		Cyclic normal subgroup Cyclic group
Odd-order cyclic group		Cyclic group Odd-order group Order of a group
Polycyclic group		Subnormal series Cyclic group Solvable group Finitely generated abelian group Noetherian group

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