Nilpotent Fuzzy Subgroups

In this paper, we introduce a new definition for nilpotent fuzzy subgroups, which is called the good nilpotent fuzzy subgroup or briefly g-nilpotent fuzzy subgroup. In fact, we prove that this definition is a good generalization of abstract nilpotent groups. For this, we show that a group G is nilpotent if and only if any fuzzy subgroup of G is a g-nilpotent fuzzy subgroup of G. In particular, we construct a nilpotent group via a g-nilpotent fuzzy subgroup. Finally, we characterize the elements of any maximal normal abelian subgroup by using a g-nilpotent fuzzy subgroup.


Introduction
Applying the concept of fuzzy sets of Zadeh [1] to group theory, Rosenfeld [2] introduced the notion of a fuzzy subgroup as early as 1971.Within a few years, it caught the imagination of algebraists like wildfire and there seems to be no end to its ramifications.With appropriate definitions in the fuzzy setting, most of the elementary results of group theory have been superseded with a startling generalized effect (see [3][4][5]).In [6] Dudek extended the concept of fuzzy sets to the set with one n-ary operation i.e., to the set G with one operation on f : G −→ G, where n ≥ 2. Such defined groupoid will be denoted by (G, f ).Moreover, he introduced the notion of a fuzzy subgroupoid of an n-ary groupoid.Specially, he proved that if every fuzzy subgroupoid µ defined on (G, f ) has the finite image, then every descending chain of subgroupoids of (G, f ) terminates at finite step.One of the important concept in the study of groups is the notion of nilpotency.In [7] Kim proposed the notion of a nilpotent fuzzy subgroup.There, he attached to a fuzzy subgroup an ascending series of subgroups of the underlying group to define nilpotency of the fuzzy subgroup.With this definition, the nilpotence of a group can be completely characterized by the nilpotence of its fuzzy subgroups.Then, in [8] Guptaa and Sarmahas, defined the commutator of a pair of fuzzy subsets of a group to generate the descending central chain of fuzzy subgroups of a given fuzzy subgroup and they proposed a new definition of a nilpotent fuzzy subgroup through its descending central chain.Specially, They proved that every Abelian (see [9]) fuzzy subgroup is nilpotent.There are many natural generalizations of the notion of a normal subgroup.One of them is subnormal subgroup.The new methods are important to guarantee some properties of the fuzzy sets; for example, see [10].In [3] Kurdachenko and et all formulated this concept for fuzzy subgroups to prove that if every fuzzy subgroup of γ is subnormal in γ with defect at most d, then γ is nilpotent ([3] Corollary 4.6 ).Finally in [11,12] Borzooei et.al. defind the notions of Engel fuzzy subgroups (subpolygroups) and investigated some related results.Now, in this paper we define the ascending series differently with Kim's definition.We then propose a definition of a nilpotent fuzzy subgroup through its ascending central series and call it g-nilpotent fuzzy subgroups.Also, we show that each g-nilpotent fuzzy subgroup is nilpotent.Moreover, we get the main results of nilpotent fuzzy subgroups with our definition.Basically this definition help us with the fuzzification of much more properties of nilpotent groups.Furthermore, we prove that for a fuzzy subgroup µ of G, {x ∈ G | µ([x, y 1 , ..., y n ]) = µ(e) f or any y 1 , ..., y n ∈ G} is equal to the n−th term of ascending series where [x, y 1 ] = x −1 y −1 1 xy 1 and [x, y 1 , ..., y n ] = [[x, y 1 , ..., y n−1 ], y n ].Therefore, we have a complete analogy concept of nilpotent groups of an abstract group.Specially, we prove that a finite maximal normal subgroup can control the g-nilpotent fuzzy subgroup and makes it finite.
Theorem 3. [14] Let µ be a fuzzy subset of a semigroup G.If Z(µ) is nonempty, then Z(µ) is a subsemigroup of G.Moreover, if G is a group, then Z(µ) is a normal subgroup of G.
We recall the notion of the ascending central series of a fuzzy subgroup and a nilpotent fuzzy subgroup of a group [14].Let µ be a fuzzy subgroup of a group G and , which implies that Z 2 (µ) is a normal subgroup of G. Similarly suppose that Z i (µ) has been defined and so [14] Let µ be a fuzzy subgroup of a group G.The ascending central series of µ is defined to be the ascending chain of normal subgroups of G as follows: The other definition is as follows [13]: Let G be a group and ). Similarly we get the result for any n ∈ N. Theorem 5. [13] Let G be a group and n ∈ N. Then (iii) Class of nilpotent groups is closed with respect to subgroups and homomorphic images.
Notation.From now on, in this paper we let G be a group.

Good Nilpotent Fuzzy Subgroups
One of the important concept in the study of groups is the notion of nilpotency.It was introduced for fuzzy subgroups, too (See [14]).Now, in this section we give a new definition of nilpotent fuzzy subgroups which is similar to one in the abstract group theory.It is a good generation of the last one.With this nilpotency we get some new main results.
Let µ be a fuzzy subgroup of G.
In the following we see that for n ∈ N, each normal subgroup Z n (µ), in which is defined by Proof.We prove it by induction on k.
This complete the proof.
Next we see that a g-nilpotent fuzzy subgroup of G makes the g-nilpotent fuzzy subgroup of G Z(µ) .For this, we need the following two Lemmas.Lemma 2. Let µ be a fuzzy subgroup of G. Then for any k ∈ N, Z k (µ) Proof.First we recall that for i ∈ N, x ∈ Z i (G) if and only if [x, y 1 , ..., y i ] = e, for any y 1 , y 2 , ..., y i ∈ G (See [13]).Hence Lemma 3. Let µ be a fuzzy subgroup of G, H = G Z(µ) , µ be a fuzzy subgroup of H and N = Z(µ).If H is nilpotent, then H N is nilpotent, too.
Proof.Let H be nilpotent of class n, that is Z n (H) = H.We will prove that there exist m ≤ n such that Z m ( H N ) = H N .For this by Theorem 5, since H N is a homomorphic image of H, we get H N is nilpotent of class at most m.Theorem 8. Let µ be a fuzzy subgroup of G and µ be a fuzzy subgroup of G Z(µ) .If µ is a g-nilpotent fuzzy subgroup of class n, then µ is a g-nilpotent fuzzy subgroup of class m, where m ≤ n.
Proof.Let µ be a g-nilpotent fuzzy subgroup of class n.Then Z n (µ) = G.Now we show that there exists ) , and similarly (put m instead of n and µ instead of µ), We now consider homomorphic images and the homomorphic pre-image of g-nilpotent fuzzy subgroups.Theorem 9. Let H be a group, f : G −→ H be an epimorphism and µ be a fuzzy subgroup of G.If µ is a g-nilpotent fuzzy subgroup, then f (µ) is a g-nilpotent fuzzy subgroup.
Proof.First, we show that f (Z i (µ)) ⊆ Z i ( f (µ)), for any i ∈ N. Let i ∈ N. Then x ∈ f (Z i (µ)) implies that x = f (u), for some u ∈ Z i (µ).Since f is epimorphism, hence for any y 1 , ..., y n ∈ H we get Theorem 10.Let H be a group, f : G −→ H be an epimorphism and ν be a fuzzy subgroup of H. Then ν is a g-nilpotent fuzzy subgroup if and only if f −1 (ν) is a g-nilpotent fuzzy subgroup.

Proof. First, we show that
Hence ν is g-nilpotent if and only if there exists nonnegative integer n such that Z n (ν) = H if and Proposition 1.Let µ and ν be two fuzzy subgroups of G such that µ ⊆ ν and µ(e) = ν(e).Then Z(µ) ⊆ Z(ν).
The proof is similar.
In the following we see a relation between nilpotency of a group and its fuzzy subgroups.
Theorem 11.G is nilpotent if and only if any fuzzy subgroup µ of G is a g-nilpotent fuzzy subgroup.

Proof. (=⇒)
Let G be nilpotent of class n and µ be a fuzzy subgroup of G. Since Z n (G) = G, it is enough to prove that for any nonnegative integer i, Z i (G) ⊆ Z i (µ).For i=0 or 1, the proof is clear.Let for i > 1, Z i (G) ⊆ Z i (µ) and x ∈ Z i+1 (G).Then for any y ∈ G, [x, y] ∈ Z i (G) ⊆ Z i (µ) and so by Lemma 4, x ∈ Z i+1 (µ).Hence Z i+1 (G) ⊆ Z i+1 (µ), for any i ≥ 0, and this implies that Z n (µ) = G.Therefore, µ is g-nilpotent.
(⇐=) Let any fuzzy subgroups of G be g-nilpotent.Suppose that fuzzy set µ on G is defined as follows: Hence, for any y, y 1 , ..., y i−1 ∈ G, [x, y, y 1 , ..., y i−1 ] = e which implies that x ∈ Z i (G).Thus by induction on i, Z i (µ) ⊆ Z i (G), for any nonnegative integer i.Now since Z i (G) ⊆ Z i (µ) for any nonnegative integer i, then Z i (µ) = Z i (G).Now by the hypotheses there exist Theorem 12. Let fuzzy subgroups µ 1 and µ 2 of G be g-nilpotent fuzzy subgroups.Then the fuzzy set µ 1 × µ 2 of G × G is a g-nilpotent fuzzy subgroup, too.Theorem 13.Let µ be a normal fuzzy subgroup of G. Then µ is a g-nilpotent fuzzy subgroup if and only if G µ is a nilpotent group.
Theorem 14.Let µ be a fuzzy subgroup of G and µ * = {x | µ(x) = µ(e)} be a normal subgroup of G.If G µ * is a nilpotent group, then µ is a g-nilpotent fuzzy subgroup.
Theorem 15.Let µ and ν be two fuzzy subgroups of G such that µ ⊆ ν and µ(e) = ν(e).If µ is a g-nilpotent fuzzy subgroup of class m , then ν is a g-nilpotent fuzzy subgroup of class n, where n ≤ m.
Proof.Let µ and ν be two fuzzy subgroups of G where µ ⊆ ν and µ(e) = ν(e).First, we show that for any , which implies that ν is g-nilpotent of class at most m.Definition 9. [4] Let µ be a fuzzy set of a set S. Then the lower level subset is Now fuzzification of µ t is the fuzzy set A µ t defined by Corollary 1.Let µ be a nilpotent fuzzy subgroup of G. Then A µ t is nilpotent too.
Proof.Let µ be a nilpotent fuzzy subgroup of G, since A µ t ⊆ µ then by Theorem 15, A µ t is nilpotent.
Lemma 7. Let µ be a g-nilpotent fuzzy subgroup of G of class n ≥ 2 and N, be a nontrivial normal subgroup of G (i.e 1 = N ¢ G).Then N ∩ Z(µ) = 1.

Lemma 4 .
Let µ be a fuzzy subgroup of G and i > 1.Then for any y
Theorem 7. Let µ be a fuzzy subgroup of G. Then µ is commutative if and only if µ is g-nilpotent fuzzy subgroup of class 1.