Cocyclic group

2020 Mathematics Subject Classification: Primary: 20A [MSN][ZBL]

A cogenerator of a group $G$ is an element $c$ such that for any homomorphism $\phi : G \rightarrow H$, then $c \not\in \ker\phi$ implies that $\phi$ is a monomorphism (injective as a map). A cocyclic group is a group with a cogenerator. An equivalent definition is that a cocyclic group has a non-trivial minimal subgroup $M$, or that the intersection $M$ of all non-trivial subgroups of $G$ is non-trivial. Each element of $M$ other than the identity is a cogenerator.

The cocyclic groups are the cyclic groups of prime power order and the quasi-cyclic groups.

The definition is dual to one characterisation of a generator $g$ of a cyclic group, that for every $\phi : H \rightarrow G$, if $g$ is in the image of $\phi$ then $\phi$ is an epimorphism (surjective as a map).

More generally, a subset $C$ of $G$ is a cogenerating set, or set of cogenerators, if every minimal subgroup of $G$ contains an element of $C$: equivalently, for any $\phi : G \rightarrow H$, then $C \cap \ker\phi = \emptyset$ implies that $\phi$ is a monomorphism.

References

• László Fuchs, "Abelian groups" Springer (2015) ISBN 978-3-319-19421-9 Zbl 06457087