$G/F$

is cyclic.

In other words, $G$ is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by').

If $F$ is a finitely generated group we say that $G$ is (finitely generated free)-by-cyclic (or (f.g. free)-by-cyclic).

References

• A. Martino and E. Ventura (2004), The Conjugacy Problem for Free-by-Cyclic Groups Preprint from the Centre de Recerca Matemàtica, Barcelona, Catalonia, Spain.