Free-by-cyclic group

In group theory, a group formula_1 is said to be free-by-cyclic if it has a free normal subgroup formula_2 such that the quotient group 
is cyclic. 
In other words, formula_1 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 formula_5 is a finitely generated group we say that formula_1 is "(finitely generated free)-by-cyclic" (or (f.g. free)-by-cyclic). 