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) − f(g) − f(h) is uniformly bounded and f(gn) = n·f(g).
- The vector space of pseudocharacters of a boundedly generated group G is finite dimensional.
Read more about this topic: Boundedly Generated Group
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)