Properties
The class of locally finite groups is closed under subgroups, quotients, and extensions (Robinson 1996, p. 429).
Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-subgroup contained in no other p-subgroups, then all maximal p-subgroups are finite and conjugate. If there are finitely many conjugates, then the number of conjugates is congruent to 1 modulo p. In fact, if every countable subgroup of a locally finite group has only countably many maximal p-subgroups, then every maximal p-subgroup of the group is conjugate (Robinson 1996, p. 429).
The class of locally finite groups behaves somewhat similarly to the class of finite groups. Much of the 1960s theory of formations and Fitting classes, as well as the older 19th century and 1930s theory of Sylow subgroups has an analogue in the theory of locally finite groups (Dixon 1994, p. v.).
Similarly to the Burnside problem, mathematicians have wondered whether every infinite group contains an infinite abelian subgroup. While this need not be true in general, a result of Philip Hall and others is that every infinite locally finite group contains an infinite abelian group. The proof of this fact in infinite group theory relies upon the Feit–Thompson theorem on the solubility of finite groups of odd order (Robinson 1996, p. 432).
Read more about this topic: Locally Finite 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 (1632–1704)
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—Ralph Waldo Emerson (1803–1882)