Hall subgroup

In mathematics, a Hall subgroup of a finite group "G" is a subgroup whose order is coprime to its index. They are named after the group theorist Philip Hall.

Definitions

A Hall divisor of an integer "n" is a divisor "d" of "n" such that"d" and "n"/"d" are coprime. The easiest way to find the Hall divisors is to write the prime factorization for the number in question and take any product of the multiplicative terms (the full power of any of the prime factors), including 0 of them for a product of 1 or all of them for a product equal to the original number. For example, to find the Hall divisors of 60, show the prime factorization is 2²·3·5 and take any product of {3,4,5}. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.

A Hall subgroup of "G" is a subgroup whose order is a Hall divisor of the order of "G". In other words, it is a subgroup whose order is coprime to its index.

If "π" is a set of primes, then a Hall "π"-subgroup is a subgroup whose order is a product of primes in "π", and whose index is not divisible by any primes in "π".

Examples

*Any Sylow subgroup of a group is a Hall subgroup.

*If "G" = "A"₅, the only simple group of order 60, then 15 and 20 are Hall divisors of the order of "G", but "G" has no subgroups of these orders.

*The simple group of order 168 has two different conjugacy classes of Hall subgroups of order 24 (though they are conjugate under an outer automorphism of "G").

*The simple group of order 660 has two Hall subgroups of order 12 that are not isomorphic.

Hall's theorem

Hall proved that if "G" is a finite solvable group and "π"is any set of primes, then "G" has a Hall "π"-subgroup, and anytwo Hall "π"-subgroups are conjugate. Moreover any subgroup whose order isa product of primes in "π" is contained in some Hall "π"-subgroup. This result can be thought of as a generalization of Sylow's Theorem to Hall subgroups, but the examples above show that such a generalization is false when the group is not solvable.

Hall's theorem can be proved by induction on the order of "G", using the fact that every finite solvable group has a normal elementary abelian subgroup.

A converse to Hall's theorem

Any finite group that has a Hall "π"-subgroup for every set of primes "π" is solvable. This is a generalization of Burnside's theorem that any group whose order is of the form "p^aq^b" for primes "p" and "q" is solvable, because Sylow's theorem implies that all Hall subgroups exist. This does not give another proof of Burnside's theorem, because Burnside's theorem is needed to prove this converse.

Sylow systems

A Sylow system is a set of Sylow "p"-subgroups "S_p" for each prime "p" such that "S_pS_q" = "S_qS_p" for all "p" and "q". If we have a Sylow system, then the subgroup generated by the groups "S_p" for "p" in "π" is a Hall "π"-subgroup. A more precise version of Hall's theorem says that any solvable group has a Sylow system, and any two Sylow systems are conjugate.

References

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

Hall–Janko graph — [ HJ as Foster graph (90 outer vertices) plus Steiner system S(3,4,10) (10 inner vertices).] In graph theory, the Hall–Janko graph, also known as the Hall Janko Wales graph, is a rank 3 strongly regular graph with parameters (100,36,14,12) and a… … Wikipedia
Hall–Janko group — In mathematics, the Hall Janko group HJ , is a finite simple sporadic group of order 604800. It is also called the second Janko group J 2, or the Hall Janko Wales group, since it was predicted by Janko and constructed by Hall and Wales. It is a… … Wikipedia
Hall's marriage theorem — In mathematics, Hall s marriage theorem is a combinatorial result that gives the condition allowing the selection of a distinct element from each of a collection of finite sets. It was proved by Philip Hall (1935). Contents 1 Definitions and … Wikipedia
Carter subgroup — In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a subgroup H that is a nilpotent group, and self normalizing. These subgroups were introduced by Roger Carter, and marked the beginning of the post… … Wikipedia
Philip Hall — Infobox Scientist name = Philip Hall caption = Philip Hall birth date = birth date|1904|4|11|df=y birth place = Hampstead, London, England death date = death date and age|1982|12|30|1904|4|11|df=y death place = Cambridge, England residence =… … Wikipedia
Focal subgroup theorem — In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in (Higman 1958) and is the first major application of the transfer according to… … Wikipedia
Omega and agemo subgroup — In mathematics, or more specifically group theory, the omega and agemo subgroups described the so called power structure of a finite p group. They were introduced in (Hall 1933) where they were used to describe a class of finite p groups whose… … Wikipedia
Normal subgroup — Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product … Wikipedia
Frattini subgroup — Hasse diagram of the lattice of subgroups of the dihedral group Dih4 In the 3 element layer are the maximal subgroups; their intersection (the F. s.) is the central element in the 5 element layer. So Dih4 has only one non generating element… … Wikipedia
Feit–Thompson theorem — In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963) Contents 1 History 2 Significance of the proof … Wikipedia

Academic Dictionaries and Encyclopedias

Hall subgroup

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Hall subgroup

Look at other dictionaries:

Share the article and excerpts

Direct link