In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a p-constrained group is a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
resembling the centralizer of an element of
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
order ''p'' in a
group of Lie type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...
over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of
characteristic ''p''. They were introduced by in order to extend some of Thompson's results about odd groups to groups with
dihedral Sylow 2-subgroups.
Definition
If a group has trivial ''p''
core
Core or cores may refer to:
Science and technology
* Core (anatomy), everything except the appendages
* Core (manufacturing), used in casting and molding
* Core (optical fiber), the signal-carrying portion of an optical fiber
* Core, the central ...
O
''p''(''G''), then it is defined to be ''p''-constrained if the ''p''-core O
''p''(''G'') contains its centralizer, or in other words if its
generalized Fitting subgroup In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the smalles ...
is a ''p''-group. More generally, if O
''p''(''G'') is non-trivial, then ''G'' is called ''p''-constrained if ''G''/O
''p''(''G'') is .
All
''p''-solvable groups are ''p''-constrained.
See also
*
''p''-stable group
*The
ZJ theorem In mathematics, George Glauberman's ZJ theorem states that if a finite group ''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd prime ''p'', then ''O'p''′(''G'')''Z''(''J''(''S'')) is a normal subgroup ...
has ''p''-constraint as one of its conditions.
References
*
*{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , title=Finite groups , url=https://www.ams.org/bookstore-getitem/item=CHEL-301-H , publisher=Chelsea Publishing Co. , location=New York , edition=2nd , isbn=978-0-8284-0301-6 , mr=569209 , year=1980
Finite groups
Properties of groups