New Properties of Algebraic Structure of Vague Groups

Because the vague (simply, V-) sets can distinguish the positive evidence from the negative one, the V-sets can represent more information than the fuzzy (simply, F-) sets. However the algebraic structure of the V-sets is simpler. The V-groups are the V-sets with V-binary operation. But, the structure of V-groups were less studied in previous works. To solve the problem, firstly, this paper studies some V-subgroups, including the commutator V-subgroups, the V-normal series for a group and their factors , and the V-subnormal series for a V-group and their factors, and the V-compostion series for a V-group and their factors. Secondly, further studying V- groups by means of V-subgroups, this paper presents and proves the six theorems: 1. The sufficient and necessary condition that V-quotient groups are simple V- groups; 2. Any two V-composition series for a group and their factors are both isomorphic; 3. The V-quotient group of V-group about a commutator V-subgroup is the Abel V-group, and any V- quotient group of V-group about V-normal subgroup containing commutator V-group is the Abel V-groups. 4.The commutator V- subgroups are the normal V-groups; 5. The solvable and simple V- groups are the cyclic V-groups with prime order, 6. The sufficient and necessary condition of the structures of the solvable V- groups. This paper reveals new properties of algebraic structure of V-groups.