Рее топтары: математикалық анықтама және қасиеттері

In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered. Unlike the Steinberg groups, the Ree groups are not given by the points of a connected reductive algebraic group defined over a finite field; in other words, there is no "Ree algebraic group" related to the Ree groups in the same way that (say) unitary groups are related to Steinberg groups. However, there are some exotic pseudo reductive algebraic groups over non perfect fields whose construction is related to the construction of Ree groups, as they use the same exotic automorphisms of Dynkin diagrams that change root lengths. defined Ree groups over infinite fields of characteristics 2 and 3. and introduced Ree groups of infinite dimensional Kac–Moody algebras.

The relation between Chevalley groups, Steinberg group, and Ree groups is roughly as follows. Given a Dynkin diagram X, Chevalley constructed a group scheme over the integers 'Z' whose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphism α of where is the algebraic closure of a finite field, such that some power of α is some power of the Frobenius endomorphism φ. The three cases are as follows:
For Chevalley groups, for some positive integer n. In this case the group of fixed points is also the group of points of X defined over a finite field. For Steinberg groups, for some positive integers m, n with m dividing n and m > 1. In this case the group of fixed points is also the group of points of a twisted (quasisplit) form of X defined over a finite field. For Ree groups, for some positive integers m, n with m not dividing n. In practice m=2 and n is odd. Ree groups are not given as the points of some connected algebraic group with values in a field. they are the fixed points of an order m=2 automorphism of a group defined over a field of order pn with n odd, and there is no corresponding field of order pn/2 (although some authors like to pretend there is in their notation for the groups).

The Ree groups of type 2B2 were first found by using a different method, and are usually called Suzuki groups. Ree noticed that they could be constructed from the groups of type B2 using a variation of the construction of Ree realized that a similar construction could be applied to the Dynkin diagrams F4 and G2, leading to two new families of finite simple groups.