Given a group G of permutations of a finite n-element set Xn and a transformation f of Xn, the G-closure 〈f : G〉 of f is the semigroup generated by all the conjugates of f by permutations in G. A semigroup S of transformations of Xn is G-normal if GS = G, where GS consists of all the permutations h of Xn such that h−1fh ∈ S for all f ∈ S. We may assume that Xn is a chain and we let POIn be the ...

Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω1, ω2, . . .) ∈ S∗ in the generators we define a partial order—called the ω-sorting order—on the set of group elements Wω ⊆ W that occur as finite subwords of ω. We show that the ω-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, t...

The main target of this article is to study about Lie Groups, Algebras. This will enrich our knowledge Algebraic properties manifolds, how Groups and Algebras are working with their properties. Finally, we have discussed an example by showing all the Algebra,Lie for a special Group Theorem has established. Dhaka Univ. J. Sci. 71(1): 82-86, 2023 (Jan)