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Groupprops β

Residually cyclic group

Definition

A group is termed residually cyclic if it satisfies the following equivalent conditions:

  1. For every non-identity element, there is a normal subgroup of finite index of the whole group not containing that element, such that the quotient group is a cyclic group.
  2. The group is isomorphic to a subdirect product of cyclic groups.
  3. The group can be embedded as a subgroup in a direct product of cyclic groups.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Cyclic group |FULL LIST, MORE INFO
Finitely generated abelian group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Residually finitely generated group |FULL LIST, MORE INFO
Abelian group |FULL LIST, MORE INFO