Properties of Infinite Groups
For an infinite group, a "small neighborhood" is taken to be a finitely generated subgroup. An infinite group is said to be locally P if every finitely generated subgroup is P. For instance, a group is locally finite if every finitely generated subgroup is finite. A group is locally soluble if every finitely generated subgroup is soluble.
Read more about this topic: Local Property
Famous quotes containing the words properties of, properties, infinite and/or groups:
“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)
“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)
“Since [man] is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the nothing from which he was made, and the infinite in which he is swallowed up.”
—Blaise Pascal (1623–1662)
“Only the groups which exclude us have magic.”
—Mason Cooley (b. 1927)