In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a p-constrained group is a
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
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 group, reductive linear algebraic group with values in a finite ...
over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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 (laboratory), a highly specialized shared research resource
* Core (manufacturing), used in casting and molding
* Core (optical fiber ...
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 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 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