p-constrained group

Short description: A finite 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 Gorenstein and Walter (1964, 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′ 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 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

• Gorenstein, D.; Walter, John H. (1964), "On the maximal subgroups of finite simple groups", Journal of Algebra 1 (2): 168–213, doi:10.1016/0021-8693(64)90032-8, ISSN 0021-8693 
• Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, https://www.ams.org/bookstore-getitem/item=CHEL-301-H