PROBABILITY THAT THE COMMUTATOR OF TWO GROUP ELEMENTS IS EQUAL TO A GIVEN ELEMENT

M. R. POURNAKI AND R. SOBHANI

Abstract. In this paper we study the probability that the commutator of two randomly chosen elements in a finite group is equal to a given element of that group. Explicit computations are obtained for groups G which |G′| is prime and G′ ≤ Z(G) as well as for groups G which |G′| is prime and G′ ∩ Z(G) = 1. This paper extends results of Rusin [see D. J. Rusin, What is the probability that two elements of a finite group commute? Pacific J. Math. 82 (1979), no. 1, 237–247].

1. Introduction

Let G be a finite group acting on the finite set Ω. Assume that Fix(G, Ω) is the set of pairs (g, ω) of G × Ω such that g.ω = ω and put p_{G}(Ω) = |Fix(G, Ω)|/|G||Ω|. Of course, this quantity is the probability that an element of G leaves an element of Ω fixed. This concept has been considered by several authors. Sherman [13] considered the case for which G is abelian and A its full automorphism group. Therefore, p_{A}(G) is the probability that an automorphism leaves an element fixed. He proved for groups of order p^{n}, p prime, p_{A}(G) ≤ 2(3/p^{2})^{n/2}. He further showed that if G_{n} is a sequence of abelian groups with |G_{n}| → ∞ then p_{A}(G_{n}) → 0.

If we assume G acts on itself by conjugation, then one obtains p_{G}(G), simply denoted by p(G), is the probability that two randomly chosen elements of G commute. Note that several other notations have been used for p(G) such as Pr(G), R(G), mc(G), and etc. An important formula for this latter probability is p(G) = k(G)/|G|, where k(G) is the number of conjugacy classes of G. The proof of this equality has been established by Gustafson [6] and the technique of the proof was used by Erdös and Turán [3]. He showed that the maximum value for k(G)/|G| for finite nonabelian groups is 5/8, so p(G) ≤ 5/8 for all such groups which is, of course, sharp. Also he showed that, with a suitable interpretation, this latter inequality remains valid for compact topological nonabelian groups. Rusin [12] again considered p(G) = k(G)/|G|, the probability that two elements of G commute, and has been obtained an explicit computation of p(G) for groups G with G′ ≤ Z(G). Also some limiting conditions were given and he classified the groups G for which p(G) is greater than 11/32.

2000 Mathematics Subject Classification. Primary: 20D99, 20F12, 20F35; Secondary: 20C15, 20D60.

Key words and phrases. Commutator, Irreducible complex character, Isoclinism.

The research of the first author was in part supported by a grant from IPM (No. 85200017).

Now let G be a finite group and g be an element of G and consider the probability p_{g}(G) that the commutator of two randomly chosen elements of G is equal to g. It is clear that p_{1}(G) is equal to the probability that two randomly chosen elements of G commute and, therefore, this latter probability is a generalization of p(G). In this paper we study p_{g}(G). There are two possibilities for groups G in which |G′| is prime: G′ ≤ Z(G) or G′ ∩ Z(G) = 1. Explicit computations for p_{g}(G) are obtained in each cases (see Sections 3 and 4). This paper extends results of Rusin [12].

2. Some computing formulas

In this section we give some formulas to compute the probability p_{g}(G) that the commutator of two randomly chosen elements in a finite group G is equal to a given element g of G. Obviously for those g’s lie in G \ G′, we have p_{g}(G) = 0. Therefore in the sequel we deal only with g’s which lie in G′. Note that there are examples of groups G of order 96 due to Guralnick [5] where p_{g}(G) = 0 even when g belongs to G′.

We now start with the following theorem which gives us a character theoretical formula for p_{g}(G). In the following Irr(G) denotes the set of all irreducible complex characters of G.

Theorem 2.1. Let G be a finite group, and let g ∈ G′. Then we have

p_{g}(G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(g)}{χ(1)}.

Proof. Let D be the set of pairs (x, y) of G × G such that [x, y] = g, where [x, y] := x^{−1}y^{−1}xy denotes the commutator of x, y. Therefore we have p_{g}(G) = |D|/|G|^{2}. On the other hand, by a theorem of Frobenius [4] the number of solutions of the equation [x, y] = g in G, i.e., |D|, is equal to |G| _{∑χ∈Irr(G)}χ(g)/χ(1). Therefore we obtain

p_{g}(G) = \frac{|D|}{|G|^{2}} = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(g)}{χ(1)},

as required. □

Let us note that if we consider g = 1 in the above theorem, then we obtain

p_{1}(G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(1)}{χ(1)} = \frac{1}{|G|} ∑_{χ∈Irr(G)}1 = \frac{|Irr(G)|}{|G|} = \frac{k(G)}{|G|},

where k(G) is the number of conjugacy classes of G. The quantity p_{1}(G) is the probability that two randomly chosen elements of G commute and so the latter equality is a well known result, as we mentioned in the previous section. Therefore Theorem 2.1 is a generalization of this well known key result.

Given a finite group G, let cd(G) = {χ(1) : χ ∈ Irr(G)} be the set of degrees of the irreducible complex characters of G. The study of the structure of a finite group G by imposing conditions on the set cd(G) has been considered in many papers in the last decades. For example, groups having just two different irreducible complex character degrees are solvable, and these groups have been thoroughly investigated (see the results by Isaacs and Passman in [10, 11], Isaacs’ book [9, Chapter 12] or Bannuscher’s papers [1, 2]). In the following theorem we give a formula to compute p_{g}(G) for finite groups with just two irreducible complex character degrees.

Theorem 2.2. Let G be a finite group such that cd(G) = {1, m}, m > 1. If g ∈ G′, then we have

p_{g}(G) = \begin{cases}\frac{1}{|G′|}(1 − \frac{1}{m^{2}} ) & if g ≠ 1, \\ \frac{1}{|G′|}(1 + \frac{|G′| − 1}{m^{2}}) & if g = 1. \end{cases}

Proof. First, we assume that g ≠ 1. By applying second orthogonality relation for g we obtain that

0 = ∑_{χ∈Irr(G)}χ(g)χ(1) = ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} χ(g)χ(1) + ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) > 1 \end{matrix}} χ(g)χ(1).

In the case that χ is linear, we have G′ ≤ Ker χ and therefore g ∈ Ker χ. Hence χ(g) = χ(1). Also the number of all linear irreducible complex characters of G is equal to |G : G′|. On the other hand, by assumption, χ(1) = m holds for each nonlinear irreducible complex character χ of G. Therefore we have

∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} χ(g)χ(1) = ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} χ(1)^{2} = ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} 1 = |G : G′|,

which implies that

|G : G′| + m ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) > 1 \end{matrix}} χ(g) = 0.

Therefore we get

∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) > 1 \end{matrix}} χ(g) = −\frac{|G : G′|}{m},

and so by Theorem 2.1 we have

p_{g}(G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(g)}{χ(1)}

= \frac{1}{|G|}( ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} \frac{χ(g)}{χ(1)} + ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) > 1 \end{matrix}} \frac{χ(g)}{χ(1)})

= \frac{1}{|G|}( ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) = 1 \end{matrix}} 1 + \frac{1}{m} ∑_{\begin{matrix}χ ∈ Irr(G) \\ χ(1) > 1 \end{matrix}} χ(g))

= \frac{1}{|G|}(|G : G′| + \frac{1}{m}( − \frac{|G : G′|}{m}))

= \frac{1}{|G′|}(1 − \frac{1}{m^{2}} ),

as required. Second, we assume that g = 1. In this case, p_{1}(G) is the probability that two randomly chosen elements of G commute, i.e., it is equal to k(G)/|G|, where k(G) is the number of conjugacy classes of G. We now have

|G| = ∑_{χ ∈ Irr(G)} χ(1)^{2} = |G : G′| + (k(G) − |G : G′|)m^{2},

so we obtain

p_{1}(G) = \frac{1}{|G′|}(1 + \frac{|G′| − 1}{m^{2}})

as required. □

It is well known that the degree of an irreducible complex character cannot exceed |G : Z(G)|^{1/2}. Finite groups for which this bound is attained are called groups of central type. In 1982 Howlett and Isaacs [8] proved that a finite group of central type must be solvable, but not necessarily nilpotent. Now in view of Theorem 2.2, we can state the following corollary.

Corollary 2.3. Let G be a finite group such that |cd(G)| = 2. If g ∈ G′ is nonidentity, then we have

p_{g}(G) ≤ \frac{1}{|G′|}(1 − \frac{1}{|G : Z(G)|}).

Also we have

p_{1}(G) ≥ \frac{1}{|G′|}(1 + \frac{|G′| − 1}{|G : Z(G)|}).

In both cases the equalities hold if and only if G is a group of central type.

Finite groups with just two irreducible complex character degrees are nonabelian, so for such groups the index of centre is greater than or equal to four. Therefore we can obtain the following corollary which gives us bounds on p_{g}(G) for a class of such groups.

Corollary 2.4. Let G be a finite group of central type such that |cd(G)| = 2. If g ∈ G′ is nonidentity, then we have p_{g}(G) ≥ 3/4|G′|. Also we have p_{1}(G) ≤ (|G′| + 3)/4|G′|.

3. Groups G which |G′| is prime and G′ ≤ Z(G)

In this section we deal with finite groups G which |G′| is prime and G′ ≤ Z(G). Note that these groups are finite nilpotent groups with nilpotency class at most two whose derived subgroups are cyclic of prime order. Using Theorem 2.2, we can obtain the following proposition which gives us an explicit formula to compute p_{g}(G) for such groups.

Proposition 3.1. Let G be a finite group such that |G′| = p, p prime, and let G′ ≤ Z(G). If g ∈ G′, then we have

p_{g}(G) = \begin{cases}\frac{1}{p}(1 − \frac{1}{|G : Z(G)|}) & if g ≠ 1, \\ \frac{1}{p}(1 + \frac{p − 1}{|G : Z(G)|}) & if g = 1. \end{cases}

Moreover, if g ∈ G′ is nonidentity, then we have p_{g}(G) ≥ 3/4p. Also we have p_{1}(G) ≤ (p + 3)/4p.

Proof. Let χ be a nonlinear irreducible complex character of G. We claim that χ vanishes outside of Z(G). In order to prove the claim suppose that g ∈ G \ Z(G). Therefore there exists an element x ∈ G such that z := [g, x] ≠ 1. Since |G′| = p is a prime, each nontrivial element of G′ is a generator for G′ and so G′ = ⟨z⟩. Now consider ρ as a ℂ-representation of G affording χ. By [9, Lemma 2.27] we have ρ(z) = εI for some ε ∈ ℂ. In the case ε = 1, we have z ∈ Ker ρ and therefore G′ ≤ Ker ρ which is a contradiction by [9, Corollary 2.23] and hypothesis. Therefore ε ≠ 1 and since χ(g) = χ(x^{−1}gx) = χ(gz) = tr ρ(gz) = tr(ρ(g)ρ(z)) = tr(ερ(g)I) = εχ(g), we obtain χ(g) = 0, so the claim holds. Corollary 2.30 of [9] now implies that χ(1)^{2} = |G : Z(G)|. Therefore G is a group of central type with just two irreducible complex character degrees, i.e., cd(G) = {1, |G : Z(G)|^{1/2}}. Now the assertion holds by Theorem 2.2 and Corollary 2.4. □

For developing our results we need the notion of isoclinism, for which we refer the reader to [7]. Note that isoclinism is an equivalence relation between groups. It is obvious that if G and H are isomorphic groups, then they are isoclinic. In what follows we state an important result which is proved in [7], that will be useful for further considerations.

Proposition 3.2. Let G be a group. Then there is a group H isoclinic to G such that Z(H) ≤ H′. If G is finite, so is any such H.

For a given finite group G, the above proposition implies that the set 𝕀𝕊𝕆(G) which is defined by {H : H is a finite group isoclinic to G such that Z(H) ≤ H′} is nonempty. Let us state the following lemma which will be useful for further considerations.

Lemma 3.3. Let G be a finite group such that G′ ≤ Z(G). Then we have 𝕀𝕊𝕆(G) = {H : H is a finite group isoclinic to G such that H′ = Z(H)}. Moreover, if we assume that |G′| = p, p prime, then {|H| : H ∈ 𝕀𝕊𝕆(G)} = {p^{n}} for some positive integer n.

Proof. If H ∈ 𝕀𝕊𝕆(G), then G and H are isoclinic, say via isoclinism (φ, ψ), and we have Z(H) ≤ H′. By assumption, G′ ≤ Z(G), therefore G/Z(G) is abelian and since G/Z(G) ≅ H/Z(H) (via φ), we obtain that H/Z(H) is abelian which implies that H′ ≤ Z(H). Hence we have H′ = Z(H). This proves the first part of the lemma. For the proof of the second part of the lemma suppose that H and K be elements of 𝕀𝕊𝕆(G). We have then |H/H′| = |H/Z(H)| = |K/Z(K)| = |K/K′|. But |H′| = |K′|, so we have |H| = |K| which implies that {|H| : H ∈ 𝕀𝕊𝕆(G)} = {l} for some positive integer l. Now suppose that |G′| = p and H be an element of 𝕀𝕊𝕆(G). If q (≠ p) is a prime divisor of |H|, nilpotency of H implies that the Sylow q-subgroup of H must be lie in Z(H) = H′ which is a contradiction since H′ is a p-group. Therefore H must be a p-group and so l has the form p^{n} for some positive integer n which completes the proof of the lemma. □

Viewing the above lemma, for the finite group G such that |G′| = p, p prime, and G′ ≤ Z(G), the positive integer n for which {|H| : H ∈ 𝕀𝕊𝕆(G)} = {p^{n}} is called isoclinic exponent of G and denoted by iso.exp(G).

We can now state the following proposition for a finite group G, in terms of isoclinic exponent of G, which gives us a formula for p_{g}(G) analogue to Proposition 3.1.

Proposition 3.4. Let G be a finite group such that |G′| = p, p prime, and let G′ ≤ Z(G). Suppose that iso.exp(G) = n. If g ∈ G′, then we have

p_{g}(G) = \begin{cases}\frac{1}{p}(1 − \frac{1}{p^{n−1}} ) & if g ≠ 1, \\ \frac{1}{p}(1 + \frac{p − 1}{p^{n−1}}) & if g = 1. \end{cases}

To prove Proposition 3.4 we need the following lemma. Note that in the sequel for a group K and an element k of that group we consider \overline{K} = K/Z(K) and \overline{k} = kZ(K), and define the map a_{K} : \overline{K} × \overline{K} ⟶ K′ by a_{K}(\overline{x}, \overline{y}) = [x, y] which is obviously well defined.

Lemma 3.5. Let G and H be two isoclinic finite groups and let (φ, ψ) be an isoclinism from G to H. If g ∈ G′, then we have p_{g}(G) = p_{ψ(g)}(H).

Proof. Since (φ, ψ) is an isoclinism from G to H, φ is an isomorphism from \overline{G} to \overline{H} and ψ is an isomorphism from G′ to H′ for which the following diagram commutes.

\overline{G} × \overline{G} ⟶^{φ×φ}  \overline{H} × \overline{H}
↓a_{G}  ↓a_{H}
G′  ⟶^{ψ}  H′

We now obtain the following identities:

|\overline{G}|^{2}p_{g}(G) = \frac{1}{|Z(G)|^{2}} |G|^{2}p_{g}(G)

= \frac{1}{|Z(G)|^{2}} |\{(x, y) ∈ G × G : [x, y] = g\}|

= \frac{1}{|Z(G)|^{2}} |\{(x, y) ∈ G × G : a_{G}(\overline{x}, \overline{y}) = g\}|

= |\{(α, β) ∈ \overline{G} × \overline{G} : a_{G}(α, β) = g\}|.

Since ψ is an isomorphism, the last quantity is equal to

|\{(α, β) ∈ \overline{G} × \overline{G} : ψ(a_{G}(α, β)) = ψ(g)\}|,

and commutativity of diagram implies that the above quantity is equal to

|\{(α, β) ∈ \overline{G} × \overline{G} : a_{H}(φ(α), φ(β)) = ψ(g)\}|.

But φ is an isomorphism, so we obtain

|\overline{G}|^{2}p_{g}(G) = |\{(α, β) ∈ \overline{G} × \overline{G} : a_{H}(φ(α), φ(β)) = ψ(g)\}|

= |\{(γ, δ) ∈ \overline{H} × \overline{H} : a_{H}(γ, δ) = ψ(g)\}|

= \frac{1}{|Z(H)|^{2}} |\{(x, y) ∈ H × H : a_{H}(\overline{x}, \overline{y}) = ψ(g)\}|

= \frac{1}{|Z(H)|^{2}} |\{(x, y) ∈ H × H : [x, y] = ψ(g)\}|

= \frac{1}{|Z(H)|^{2}} |H|^{2}p_{ψ(g)}(H)

= |\overline{H}|^{2}p_{ψ(g)}(H).

But \overline{G} and \overline{H} are isomorphic (via φ), hence |\overline{G}| = |\overline{H}| and the equality p_{g}(G) = p_{ψ(g)}(H) follows as required. □

Proof of Proposition 3.4. Chose an element of 𝕀𝕊𝕆(G), say H. Therefore G and H are isoclinic (say via (φ, ψ)) and we have p^{n} = |H|. Since G′ ≤ Z(G) and H ∈ 𝕀𝕊𝕆(G), Lemma 3.3 implies that H′ = Z(H). Now by using Lemma 3.5 and Proposition 3.1 we have

p_{g}(G) = p_{ψ(g)}(H)

= \begin{cases}\frac{1}{p}(1 − \frac{1}{|H : Z(H)|}) & if g ≠ 1 \\ \frac{1}{p}(1 + \frac{p − 1}{|H : Z(H)|}) & if g = 1 \end{cases}

= \begin{cases}\frac{1}{p}(1 − \frac{1}{p^{n−1}} ) & if g ≠ 1 \\ \frac{1}{p}(1 + \frac{p − 1}{p^{n−1}}) & if g = 1 \end{cases}

as required. □

4. Groups G which |G′| is prime and G′ ∩ Z(G) = 1

In this section we deal with finite groups G which |G′| is prime and G′ ∩ Z(G) = 1. We give an explicit formula for p_{g}(G) for such groups. Following we state an important result which is proved in [12], Proposition 5, that will be useful for further considerations.

Proposition 4.1. Let G be a finite group such that |G′| = p, p prime, and let Z(G) = 1. Then G = ⟨a, b : a^{p} = b^{n} = 1, bab^{−1} = a^{r}⟩, where n|(p − 1) and r^{j} ≡ 1 (mod p) if and only if n|j.

Note that the groups in the above proposition are known as metacyclic groups. The following proposition gives us a formula for p_{g}(G) for these groups.

Proposition 4.2. Let n and r be positive integers and let p be a prime number for which n|(p − 1) and r^{j} ≡ 1 (mod p) if and only if n|j. Suppose that G = ⟨a, b : a^{p} = b^{n} = 1, bab^{−1} = a^{r}⟩. If g ∈ G′, then we have

p_{g}(G) = \begin{cases}\frac{n^{2} − 1}{pn^{2}} & if g ≠ 1, \\ \frac{n^{2} + p − 1}{pn^{2}} & if g = 1. \end{cases}

Proof. It is easy to see that |G′| = p and cd(G) = {1, n}. Now the assertion holds by Theorem 2.2. □

We need the following lemma which is proved in [7], to find a formula for p_{g}(G) for finite groups G which |G′| is prime and G′ ∩ Z(G) = 1.

Lemma 4.3. Let G be a finite group and N be a normal subgroup of G for which G′ ∩ N = 1. Then G is isoclinic to G/N.

Now consider a finite group G such that |G′| = p is a prime and G′ ∩ Z(G) = 1. By the above lemma, G is isoclinic to \overline{G}. Since (\overline{G})′ = G′Z(G)/Z(G) = (G′ × Z(G))/Z(G) ≅ G′, \overline{G} has derived subgroup of order p. On the other hand, if we consider Z(\overline{G}) = H/Z(G) for some H, Z(G) ≤ H ≤ G, then [G, H] ≤ G′ ∩ Z(G) = 1 implying H = Z(G) and we have Z(\overline{G}) = 1. Therefore by Proposition 4.1, there is a positive integer n depends only to G for which \overline{G} = ⟨a, b : a^{p} = b^{n} = 1, bab^{−1} = a^{r}⟩. The number n here called invariant number of G and denoted by i(G).

Finally we can state our formula which we obtained by using definition of invariant number, Lemma 3.5 and Proposition 4.2.

Proposition 4.4. Let G be a finite group such that |G′| = p, p prime, and G′∩Z(G) = 1. Suppose that i(G) = n. If g ∈ G′, then we have

p_{g}(G) = \begin{cases}\frac{n^{2} − 1}{pn^{2}} & if g ≠ 1, \\ \frac{n^{2} + p − 1}{pn^{2}} & if g = 1. \end{cases}

5. Upper bounds for p_{g}(G)

In this section we give upper bounds for p_{g}(G). The following proposition gives us an upper bound for this quantity depending only on the number of conjugacy classes of G.

Proposition 5.1. Let G be a finite group, and let g ∈ G′. Then p_{g}(G) ≤ k(G)/|G|, where k(G) is the number of conjugacy classes of G. Moreover, the equality holds if and only if g = 1.

Proof. By [9, Lemma 2.15(c)] we have |χ(g)| ≤ χ(1) for each χ ∈ Irr(G). Therefore, Theorem 2.1 now implies that

p_{g}(G) = \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{χ(g)}{χ(1)} ≤ \frac{1}{|G|} ∑_{χ∈Irr(G)} \frac{|χ(g)|}{χ(1)} ≤ \frac{1}{|G|} ∑_{χ∈Irr(G)}1 = \frac{|Irr(G)|}{|G|} = \frac{k(G)}{|G|}.

It is obvious that the equality holds if and only if |χ(g)| = χ(1) for each χ ∈ Irr(G) or equivalently g = 1. □

Proposition 5.2. Let G be a finite group, and let g be a nonidentity element of G′. Then we have p_{g}(G) < 1/2. Also we have p_{1}(G) ≤ 5/8.

Proof. First, we assume that g ≠ 1. If p_{g}(G) ≥ 1/2, then by Proposition 5.1 we have p_{1}(G) = k(G)/|G| > p_{g}(G) ≥ 1/2 or p_{1}(G) > 1/2. Therefore we have |G| = 2^{2s+1}, |G′| = 2 and \overline{G} ≅ ℤ_{2}^{2s} for some positive integer s. But in this case we have p_{g}(G) = (1 − 1/2^{s})/2 < 1/2 which is a contradiction. Therefore p_{g}(G) < 1/2 as required. The second part is well known. □

Proposition 5.3. For any ε ∈ ℝ with ε > 0, there exists a finite group G and g ∈ G such that 1/2 − ε < p_{g}(G) < 1/2.

Proof. Let s be a positive integer such that s > −log_{2} 2ε. Consider a finite group G of order 2^{2s+1} such that |G′| = 2 and \overline{G} ≅ ℤ_{2}^{2s} and let g be a nonidentity element of G′. Then 1/2 − ε < p_{g}(G) = (1 − 1/2^{s})/2 < 1/2 as required. □

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M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir
URL: http://math.ipm.ac.ir/pournaki/

R. Sobhani, Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 85145, Isfahan, Iran, and Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

E-mail address: sbnreza@math.iut.ac.ir