**Properties**

- Bounded generation is unaffected by passing to a subgroup of finite index: if
*H*is a finite index subgroup of*G*then*G*is boundedly generated if and only if*H*is boundedly generated. - Any quotient group of a boundedly generated group is also boundedly generated.
- A finitely generated periodic group must be
*finite*if it is boundedly generated; equivalently, an*infinite*finitely generated periodic group is not boundedly generated.

A *pseudocharacter* on a discrete group *G* is defined to be a real-valued function *f* on a *G* such that

*f*(*gh*) โ*f*(*g*) โ*f*(*h*) is uniformly bounded and*f*(*g**n*) =*n*ยท*f*(*g*).

- The vector space of pseudocharacters of a boundedly generated group
*G*is finite dimensional.

Read more about this topic: Boundedly Generated Group

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