View source for Cocyclic group

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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]] ([[injection|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 group]]s of prime power order and the [[quasi-cyclic group]]s.

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]] ([[surjection|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}}