Free-by-cyclic group

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

is cyclic.

In other words, 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 is a finitely generated group we say that is (finitely generated free)-by-cyclic (or (f.g. free)-by-cyclic).