Properties
Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle.
Theorem. Let F be a countably (topologically) generated profinite group. Then
- F is projective if and only if any finite embedding problem for F is solvable.
- F is free of countable rank if and only if any finite embedding problem for F is properly solvable.
Read more about this topic: Embedding Problem
Famous quotes containing the word properties:
“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 (1803–1882)
“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)