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 order ''p'' in a
group of Lie type over a
finite field 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 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