p-constrained group
{{Short description|A finite group}}
{{lowercase|p-constrained group}}
In mathematics, a p-constrained group is a finite group 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 {{harvs|txt|last1=Gorenstein|last2=Walter|year=1964|loc=p.169}} 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{{prime}} core Op{{prime}}(G), then it is defined to be p-constrained if the p-core Op(G) contains its centralizer, or in other words if its generalized Fitting subgroup is a p-group. More generally, if Op{{prime}}(G) is non-trivial, then G is called p-constrained if G/Op{{prime}}(G) is {{nowrap|p-constrained}}.
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 | last2=Walter | first2=John H. | title=On the maximal subgroups of finite simple groups | doi=10.1016/0021-8693(64)90032-8 |mr=0172917 | year=1964 | journal=Journal of Algebra | issn=0021-8693 | volume=1 | issue=2 | pages=168–213| doi-access= }}
- {{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}}