The paper uses the tools about algebraic number theory to find a class of quartic algebraic integer ±p~(1/2)±q~(1/2),then,it is determined and proved that their minimal polynomial is [x2-(p+q)]2-4pq,and in their normal closure,there are four real inserts and no complex inserts.

In this paper,according to the meaning of the normal closure and the operating characteristic of permutation,the generation algorithm of the normal closure of the symmetric group subset is designed,and the corresponding computer program is composed by using C language.

Secondly, let H be a closed normal subgroup ofG ,we discuss the relationship between B ( H ), B ( G / H )and B ( G ),but we can’t obtain the result that B (G / H ) ??j→B (G ) ?h?→B ( H)is also a short exact sequence.

The authors study the fundamental sequence of closed regular fuzzy matroids,and find some results concerning fundamental sequence:⑴some sufficient conditions of the fundamental sequence of closed regular fuzzy matroids;⑵some properties of its fuzzy dual matroids;⑶some properties of fuzzy bases of closed regular fuzzy matroids;⑷two sufficient conditions of regular fuz.

The exchange theorem of bases of closed regular fuzzy matroids is obtained with the help of "the exchange theorem of bases" of crisp matroids.

Basing on the existing theory of crisp matroids and fuzzy matroids, the paper studies mainly closed regular fuzzy matroids including its sub-matroid when it’s good-base; the exchange of fuzzy bases; the relations between the fuzzy bases and the fuzzy circuits; defining the base-graph and studying its properties.

Methods L-closure spaces,some special maps between them and normal L-closure spaces are defined.

Some algebraic groups are discussed by looking into the lattice of their closed connected normal subgroups.

There are particular relations between the closed connected normal subgroups of algebraic groups and the ideals of Lie Algebras.