Free-by-cyclic group

In group theory, a group ${\displaystyle G}$ is said to be free-by-cyclic if it has a free normal subgroup ${\displaystyle F}$ such that the quotient group

${\displaystyle G/F}$

is cyclic.

In other words, ${\displaystyle G}$ 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 ${\displaystyle F}$ is a finitely generated group we say that ${\displaystyle G}$ is (finitely generated free)-by-cyclic (or (f.g. free)-by-cyclic).