The aim is to show that a definable group in such fields must be closely related to the rational points of an algebraic group and to investigate the structure fixed pointwise by a generic

automorphism in a generic differential difference field.

Then is it true that for any KX-negative extremal ray R [subset] [bar.NE](X) of divisorial type, there exists a positive integer k such that [([f.sup.k]).sub.*](R) = R for the

automorphism [([f.sup.k]).sub.*] : [N.sub.1](X) [equivalent] [N.sub.1](X) induced from the k-th power [f.sup.k] = f [??] ...

The Nakayama

automorphism has not been studied explicitly for skew PBW extensions.

Nakayama

automorphism is a distinguished k-algebra

automorphism of a Frobenius algebra A which measures how far A is from being a symmetric algebra, where k is a fixed field.

(iii) The Dunkl transform f [right arrow] [??] is a topological

automorphism on S([R.sup.d]).

g--the inner

automorphism of G, generated by an element g [member of] G;

[[bar.f].sub.n] is a braid-like

automorphism of [F(k)/F[(k).sub.n]] that is (i) it sends the class of each generator [[alpha].sub.i] into a conjugate of itself and (ii) it sends the class of the product [[alpha].sub.1] [[alpha].sub.2] ...

We also give an explicit form of the

automorphism group of G(H, f).

Graph r is called a vertex-transitive graph, if, for any x, y [member of] V, there is some [pi] in Aut([GAMMA]), the

automorphism group of [bar.[GAMMA]], such that [pi](x) = y.

Bhutani [2] discussed

automorphism of fuzzy graphs.

In addition the group of all

automorphisms of a free group F, denoted by Aut(F) , is generated by a regular Nielsen transformation between two basis of F, and each regular Nielsen transformation between two basis of F defines an

automorphism of F, see ([8], Korollar 2.10).

Note that an

automorphism [phi] of a field F is a bijection [phi] : F [right arrow] F such that [phi](a + b) = [phi](a) + [phi](b) and [phi](ab) = [phi](a)[phi}(b) for all a, b [member of] F.