**Free Groups Are Not Boundedly Generated**

Several authors have stated in the mathematical literature that it is obvious that finitely generated free groups are not boundedly generated. This section contains various obvious and less obvious ways of proving this. Some of the methods, which touch on bounded cohomology, are important because they are geometric rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups.

Since for any *n* ≥ 2, the free group on 2 generators *F*_{2} contains the free group on *n* generators *F*_{n} as a subgroup of finite index (in fact *n* – 1), once one non-cyclic free group on finitely many generators is known to be not boundedly generated, this will be true for all of them. Similarly, since *SL*_{2}(**Z**) contains *F*_{2} as a subgroup of index 12, it is enough to consider *SL*_{2}(**Z**). In other words, to show that no *F*_{n} with *n* ≥ 2 has bounded generation, it is sufficient to prove this for one of them or even just for *SL*_{2}(**Z**) .

Read more about this topic: Boundedly Generated Group

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