模糊水平子群与正规模糊水平子群

Fuzzy Level Subgroup and Normal Fuzzy Level Subgrpup

P. Sivaramakrishna Das~[1]和吴望名~[2]分别引入了Fuzzy水平子群和正规Fuzzy子群等概念及一些基本性质。本文在[1]和[2]的基础上引入正规水平子群,并讨论水平予群和正现水平子群的个数对群所产生的影响。

P. Sivaramakrishna Das and Wu Wangming introduced respectively the concepts of fuzzy level subgroups and the normal fuzzy subgroup, and discussed its elementary properties.In this paper, we define the concept of normal fuzzy level subgroup and obtain mainly on the bases of the paper [1] and [2] as follows.Theorem 1 If the number of level subgroups of each fuzzy subgroup on group G is finite then G is periodic.Theorem 2 Let group G have at least 3 elements. If the number of level subgroups of each fuzzy subgroup on G is at most 3 then we hold(1) The order of each element in G is prime, or product of two prime numbers(2) G is generated by two elements of which order is prime, or G is a cyclic group of order pq, or G is a cyclic group of prime order, where p and q are prime. In parcticular we haveTheorem 3 Let G be an abelian group which has at least 3 elements. Then the number of level subgroups of each fuzzy subgroup on G is at most 3 if and only if G is a cyclic group of prime order, or a cyclic group of order pq, or a non-cyclic group of order p^2. Where p and q are prime.Theorem 4 The number of normal level subgroups of each normal fuzzy group on group G is at most 2 if and only if G is simple.