Key Properties
Since the presentation for contains all the generators and relations from the presentation for, there is a natural homomorphism, induced by the identification of generators, which takes to . Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of into . A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.
Read more about this topic: HNN Extension
Famous quotes containing the words key and/or properties:
“Woe unto you, lawyers! for ye have taken away the key of knowledge: ye entered not in yourselves, and them that were entering in ye hindered.”
—Bible: New Testament Jesus, in Luke, 11:52.
“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 (1632–1704)