- Locally finite group
In
mathematics , in the field ofgroup theory , a locally finite group is a type of group that can be studied in ways analogous to afinite group .Sylow subgroup s,Carter subgroup s, andabelian subgroup s of locally finite groups have been studied.Definition and first consequences
A locally finite group is a group for which every finitely generated
subgroup is finite.Since the
cyclic subgroup s of a locally finite group are finite, every element has finite order, and so the group is periodic.Examples and non-examples
Examples:
* Every finite group is locally finite
* Every infinitedirect sum of finite groups is locally finite harv|Robinson|1996|p=443
* ThePrüfer group s are locally finite abelian groups
* Every periodic solvable group is locally finite harv|Dixon|1994|loc=Prop. 1.1.5.
* Everysubgroup of a locally finite group is locally finite. If "G" is a group and "S" is a subgroup of "G" and "F" is a finite subset of "S", the subgroup generated by "F" cannot be an infinite subset of "S" for then it would be and infinite subset of "G" contradicting the fact that "G" is locally finite.
* If G is not a locally finite group, then it is possible that there is a locally finite subgroup of G; in particular, thetrivial group .Non-examples:
* No group with an element of infinite order is a locally finite group
* No nontrivialfree group is locally finite
* ATarski monster group is periodic, but not locally finite.Properties
The class of locally finite groups is closed under subgroups, quotients, and extensions harv|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 harv|Robinson|1996|p=429.The class of locally finite groups behaves somewhat similarly to the class of finite groups. Much of the 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 harv|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 ofPhilip 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 theFeit-Thompson theorem on the solubility of finite groups of odd order harv|Robinson|1996|p=432.References
*Citation | last1=Dixon | first1=Martyn R. | title=Sylow theory, formations and Fitting classes in locally finite groups | publisher=World Scientific Publishing Co. Inc. | location=River Edge, NJ | series=Series in Algebra | isbn=9789810217952 | id=MathSciNet | id = 1313499 | year=1994 | volume=2
*External links
*springer|id=L/l060410|author=A.L. Shmel'kin
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