Free by cyclic group

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

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).