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 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 Op′(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′(G) is non-trivial, then G is called p-constrained if G/ Op′(G) is p-constrained.
All p-solvable groups are p-constrained.
gollark: It seems vaguely like complaining about food having chemicals in it, which would be very stupid, except there is apparently decent evidence of "processed" things being bad, whatever that means.
gollark: It kind of annoys me when people complain about "processed" foods because they never seem to actually explain what "processing" does which is so bad or what even counts as "processed".
gollark: Also, you apparently didn't hide anyone else's faces. That's probably impressive, though? I mean, I don't have context for such numbers, but they seem big.
gollark: I checked on the internet™, and apparently there are something like 10 combat-sports places in [somewhat nearby city I go to school in]. I'm sort of wondering if there's some local history I've missed. [nearby city] is still something like 25 minutes to travel to from where I am, which is annoying, and there don't seem to be any nearer ones.
gollark: > I'd say exercise is pretty fun if it's combat sportsI should probably try that (those?) when stuff reopens here.
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: 168–213, doi:10.1016/0021-8693(64)90032-8, ISSN 0021-8693, MR 0172917
- Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 0569209
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