On the Basic k-nacci Sequences in Finite Groups

Abstract

We define the basic k-nacci sequences and the basic periods of these sequences in finite groups, then we obtain the basic periods of the basic k-nacci sequences and the periods of the k-nacci sequences in symmetric group [subscript]S4[/subscript] , its subgroups, and binary polyhedral groups which related with these groups.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ömür Deveci 1, 2 and Erdal Karaduman 1, 2

Recommended by Carlo Piccardi

1, Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100 Kars, Turkey
2, Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey

Received 13 December 2010; Accepted 16 March 2011

1. Introduction

The study of Fibonacci sequences in groups began with the earlier work of Wall [1], where the ordinary Fibonacci sequences in cyclic groups were investigated. In the mid-eighties, Wilcox extended the problem to Abelian groups [2]. The theory is expanded to some finite simple groups by Campbell et al. [3]. There, they defined the Fibonacci length of the Fibonacci orbit and the basic Fibonacci length of the basic Fibonacci orbit in a 2-generator group. The concept of Fibonacci length for more than two generators has also been considered; see, for example, [4, 5]. Also, the theory has been expanded to the nilpotent groups; see, for example, [6, 7]. Other works on Fibonacci length are discussed in, for example, [8-10]. Knox proved that the periods of k -nacci (k- step Fibonacci) sequences in dihedral groups were equal to 2k+2 [11]. Deveci, Karaduman, and Campbell examined the period of the k -nacci sequences in some finite binary polyhedral groups in [12]. Recently, k -nacci sequences have been investigated; see, for example, [13, 14].

This paper defines the basic k -nacci sequences and the periods of these sequences in finite groups and discusses the basic periods of the basic k -nacci sequences and the periods of the k -nacci sequences in the symmetric group _S4 , alternating group _A4 , _D2 four-group, and binary polyhedral groups ...2,3,4... and ...2,3,3... with related _S4 and _A4 , respectively. We consider the groups _S4 , _A4 , ...2,3,4... , and ...2,3,3... both as 2-generator and as 3-generator groups.

A k-nacci sequence in a finite group is a sequence of group elements _x0 ,_x1 ,_x2 ,...,_xn ,... for which, given an initial (seed) set _x0 ,_x1 ,_x2 ,...,_xj-1 , each element is defined by [figure omitted; refer to PDF] We also require that the initial elements of the sequence _x0 ,_x1 ,_x2 ,...,_xj-1 generate the group, thus forcing the k- nacci sequence to reflect the structure of the group. The k -nacci sequence of a group G generated by _x0 ,_x1 ,_x2 ,...,_xj-1 is denoted by _Fk (G;_x0 ,_x1 ,...,_xj-1 ) [11].

A sequence of group elements is periodic if, after a certain point, it consists only of repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence . For example, the sequence a,b,c,d,e,b,c,d,e,b,c,d,e,... is periodic after the initial element a and has period 4. A sequence of group elements is simply periodic with period k if the first k elements in the sequence form a repeating subsequence. For example, the sequence a,b,c,d,e,f,a,b,c,d,e,f,a,b,c,d,e,f,... is simply periodic with period 6. In [11], Knox had denoted the period of a k -nacci sequence _Fk (G;_x0 ,_x1 ,...,_xj-1 ) by _Pk (G;_x0 ,_x1 ,...,_xj-1 ) .

Definition 1.1.

For a finitely generated group G=...A... , where A={_a1 ,_a2 ,...,_an } , the sequence _xi =_ai+1 , 0≤i≤n-1 , _xi+n =^∏j=1n_xi+j-1 , i≥0 is called the Fibonacci orbit of G with respect to the generating set A , denoted as _FA (G) [4].

Definition 1.2.

If _FA (G) is simply periodic, then the period of the sequence is called the Fibonacci length of G with respect to generating set A , written, LE_NA (G) [4].

Notice that the orbit of a k- generated group is a k -nacci sequence.

Let G be a finite j -generator group, and let X be the subset of G×G×G...×G...j such that (_x0 ,_x1 ,...,_xj-1 )∈X if and only if G is generated by _x0 , _x1 , ..., _xj-1 . We call (_x0 ,_x1 ,...,_xj-1 ) a generating j -tuple for G .

2. Basic Period of Basic k -nacci Sequence

To examine the concept more fully, we study the action of automorphism group Aut G of G on X and on the k -nacci sequences _Fk (G:_x0 ,_x1 ,..._xj-1 ) , (_x0 ,_x1 ,...,_xj-1 )∈X . Now, Aut G consists of all isomorphism θ:G[arrow right]G and if θ∈ Aut G and (_x0 ,_x1 ,...,_xj-1 )∈X , then (_x0 θ,_x1 θ,...,_xj-1 θ)∈X .

For a subset A⊆G and θ∈ Aut G , the image of A under θ is [figure omitted; refer to PDF]

Definition 2.1.

For a generating pair (x,y)∈X , the basic Fibonacci orbit _F¯x,y of the basic length m is defined by the sequence {_bi } of elements of G such that [figure omitted; refer to PDF] where m≥1 is the least integer with [figure omitted; refer to PDF] for some θ∈ Aut G . Since _bm ,_bm+1 generate G , it follows that θ is uniquely determined. For more information, see [3].

Lemma 2.2.

Let (_x0 ,_x1 ,...,_xj-1 )∈X and let θ∈ Aut G , then (_Fk (G:_x0 ,_x1 ,..._xj-1 ))θ=_Fk (G:_x0 θ,_x1 θ,..._xj-1 θ) .

Proof.

Let _Fk (G:_x0 ,_x1 ,..._xj-1 )={_bi } . The result is obvious since {_bi }θ={_bi θ} and [figure omitted; refer to PDF] Each generating j -tuple (_x0 ,_x1 ,...,_xj-1 )∈X maps to |AutG| distinct elements of X under the action of elements of Aut G . Hence, there are [figure omitted; refer to PDF] (where |X| means the number of elements of X ) nonisomorphic generating j -tuples for G . The notation _dj (G) was introduced in [15].

Suppose that ω elements of Aut G map _Fk (G:_x0 ,_x1 ,..._xj-1 ) into itself, then there are |AutG|/ω distinct k -nacci sequences _Fk (G:_x0 θ,_x1 θ,..._xj-1 θ) for θ∈ Aut G .

Definition 2.3.

For a j -tuple (_x0 ,_x1 ,...,_xj-1 )∈X , the basic k -nacci sequence _F¯k (G:_x0 ,_x1 ,..._xj-1 ) of the basic period m is a sequence of group elements _b0 ,_b1 ,_b2 ,...,_bn ,... for which, given an initial (seed) set _b0 =_x0 , _b1 =_x1 , _b2 =_x2 ,...,_bj-1 =_xj-1 , each element is defined by [figure omitted; refer to PDF] where m≥1 is the least integer with [figure omitted; refer to PDF] for some θ∈ Aut G . Since G is a finite j -generator group and _bm , _bm+1 , ... , _bm+j-1 generate G , it follows that θ is uniquely determined. The basic k -nacci sequence _F¯k (G:_x0 ,_x1 ,..._xj-1 ) is finite containing m element.

In this paper, we denote the basic period of the basic k -nacci sequence _F¯k (G:_x0 ,_x1 ,..._xj-1 ) by B_Pk (G;_x0 ,_x1 ,...,_xj-1 ) .

From the definitions, it is clear that the periods of the k -nacci sequences and the basic k -nacci sequences in a finite group depend on the chosen generating set and the order of the generating elements.

Theorem 2.4.

Let G be a finite group and (_x0 ,_x1 ,...,_xj-1 )∈X . If _Pk (G;_x0 ,_x1 ,...,_xj-1 )=n and B_Pk (G;_x0 ,_x1 ,...,_xj-1 )=m , then m divides n , and there are n/m elements of Aut G which map _Fk (G:_x0 ,_x1 ,..._xj-1 ) into itself.

Proof.

We have n=mλ where λ is the order of automorphism θ∈ Aut G since [figure omitted; refer to PDF] and B_Pk (G;_x0 ,_x1 ,...,_xj-1 )=B_Pk (G;_x0 θ,_x1 θ,...,_xj-1 θ) . Clearly, 1, θ, ^θ2 , ..., ^θλ-1 map _Fk (G:_x0 ,_x1 ,..._xj-1 ) into itself.

3. Applications

Definition 3.1.

The polyhedral group (l,m,n) for l,m,n>1 is defined by the presentation [figure omitted; refer to PDF] or [figure omitted; refer to PDF] The polyhedral group (l,m,n) is finite if and only if the number [figure omitted; refer to PDF] is positive, that is, in the cases (2,2,n) , (2,3,3) , (2,3,4) , and (2,3,5) . Its order is 2lmn/μ . _A4 , _S4 , and _A5 are the groups (2,3,3) , (2,3,4) , and (2,3,5) , respectively. Also, the groups _A4 , _S4 , and _A5 being isomorphic to the groups of rotations of the regular tetrahedron, octahedron, and icosahedron. Using Tietze transformations, we may show that (l,m,n)[congruent with](m,n,l)[congruent with](n,l,m) . For more information on these groups, see [16] and [17, pp. 67-68].

Definition 3.2.

The binary polyhedral group ...l,m,n... , for l,m,n>1 , is defined by the presentation [figure omitted; refer to PDF] or [figure omitted; refer to PDF] The binary polyhedral group ...l,m,n... is finite if and only if the number k=lmn(1/l+1/m+1/n-1)=mn+nl+lm-lmn is positive. Its order is 4lmn/k .

For more information on these groups, see [17, pp. 68-71].

Definition 3.3.

Let ^fn(k) denote the n th member of the k -step Fibonacci sequence defined as [figure omitted; refer to PDF] with boundary conditions ^fi(k) =0 for 1≤i<k and ^fk(k) =1 . Reducing this sequence by a modulo m , we can get a repeating sequence, which we denote by [figure omitted; refer to PDF] where ^fi(k,m) =^fi(k) (mod m) . We then have that (^f1(k,m) , ^f2(k,m) ,..., ^fk(k,m) )=(0,0,...0,1) , and it has the same recurrence relation as in (3.6) [18].

Theorem 3.4 ( f(k,m) is a periodic sequence [18]).

Let _hk (m) denote the smallest period of f(k,m) , called the period of f(k,m) or the wall number of the k-step Fibonacci sequence modulo m .

Theorem 3.5.

The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the group _S4 are as follows.

if the group is defined by the presentation _S4 =...x,y,z:^x2 =^y3 =^z4 =xyz=e... , then

(i) if k=2,_P2 (_S4 ;y,z,x)=18 and B_P2 (_S4 ;y,z,x)=9 ,

(ii) if k>2,_Pk (_S4 ;x,y,z)=6k+6 and B_Pk (_S4 ;x,y,z)=3k+3 .

If _S4 has the presentation _S4 =...x,y:^x2 =^y3 =^(xy)4 =e... , then

(i[variant prime]): if k=2,_P2 (_S4 ;x,y)=18 and B_P2 (_S4 ;x,y)=9 ,

(ii[variant prime]): if k>2 , _Pk (_S4 ;x,y)=6k+6 and B_Pk (_S4 ;x,y)=3k+3 .

Proof.

Firstly, let us consider the 3-generator case. We first note that |x|=2 , |y|=3 , and |z|=4 (where |x| means the order of x ).

(i) If k=2 , we have the sequence for the generating triple ( y , z , x ),

[figure omitted; refer to PDF] which has period 18 and the basic period 9 since xθ=x , yθ=xyx , and zθ=x^y2 , where θ is the inner automorphism induced by conjugation by x .

(ii) If k=3 , we have the sequence for the generating triple ( x , y , z ),

[figure omitted; refer to PDF] which has period 24 and the basic period 12 since xθ=x , yθ=^y2 , and zθ=yx where θ is an outer automorphism of order 2.

If k≥4 , the first k elements of sequence for the generating triple ( x , y , z ) are [figure omitted; refer to PDF] Thus, using the above information, sequence reduces to [figure omitted; refer to PDF] where _xj =e for 3≤j≤k-1 . Thus, [figure omitted; refer to PDF] where _xj =e for k+5≤j≤2k+1 , 2k+6≤j≤3k+2 , 3k+6≤j≤4k+3 , 4k+8≤j≤5k+4 , and 5k+9≤j≤6k+5 .

We also have [figure omitted; refer to PDF] Since the elements succeeding _x6k+6 , _x6k+7 , and _x6k+8 depend on x , y , and z for their values, the cycle begins again with the 6k+^6th element, that is, _x0 =_x6k+6 , _x1 =_x6k+7 , _x2 =_x6k+8 , ... . Thus, _Pk (_S4 ;x,y,z)=6k+6 .

It is easy to see from the above sequence that [figure omitted; refer to PDF] B_Pk (_S4 ;x,y,z)=3k+3 since xθ=x , yθ=^y2 , and zθ=yx where θ is an outer automorphism of order 2.

Secondly, let us consider the 2-generator case. We first note that |x|=2 , |y|=3 , and |xy|=4 .

(i[variant prime]): If k=2,_P2 (_S4 ;x,y)=18 and B_P2 (_S4 ;x,y)=9 since xθ=x and yθ=xyx where θ is the inner automorphism induced by conjugation by x .

(ii[variant prime]): If k>2 , _Pk (_S4 ;x,y)=6k+6 and B_Pk (_S4 ;x,y)=3k+3 since xθ=x and yθ=^y2 where θ is an outer automorphism of order 2.

The proofs are similar to above and are omitted.

Theorem 3.6.

The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the binary polyhedral group ...2,3,4... are as follows.

If the group is defined by the presentation ...2,3,4...=...x,y,z:^x2 =^y3 =^z4 =xyz... , then

(i) if k=2,_P2 (...2,3,4...;y,z,x)=18 and B_P2 (...2,3,4...;y,z,x)=9 ,

(ii) if k>2,_Pk (...2,3,4...;x,y,z)=6k+6 and B_Pk (...2,3,4...;x,y,z)=6k+6 .

If the group is defined by the presentation ...2,3,4...=...x,y:^x2 =^y3 =^(xy)4 ... , then

(i[variant prime]): if k=2,_P2 (...2,3,4...;x,y)=18 and B_P2 (...2,3,4...;x,y)=9 ,

(ii[variant prime]): if k>2 , _Pk (...2,3,4...;x,y)=6k+6 and B_Pk (...2,3,4...;x,y)=6k+6 .

Proof.

Firstly, let us consider the 2-generator case. We first note that |x|=4 , |y|=6 , and |xy|=8 .

(i[variant prime]): If k=2 , we have the sequence for the generating pair ( x , y ),

[figure omitted; refer to PDF] which has period 18 and the basic period 9 since xθ=^x3 and yθ=^x3 yx where θ is a outer automorphism of order 2.

(ii[variant prime]): If k=3 , we have the sequence for the generating pair ( x , y ),

[figure omitted; refer to PDF] which has period 24 and the basic period 24 since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by ^x2 .

If k=4 , we have the sequence for the generating pair ( x , y ), [figure omitted; refer to PDF] which has period 30 and the basic period 30 since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by ^x2 .

If k≥5 , the first k elements of sequence for the generating pair ( x , y ) are [figure omitted; refer to PDF] Thus, using the above information, sequence reduces to [figure omitted; refer to PDF] where _xj =e for 5≤j≤k-1 . Thus, [figure omitted; refer to PDF] [figure omitted; refer to PDF] where _xj =e for k+5≤j≤2k+1 , 2k+6≤j≤3k+2 , 3k+8≤j≤4k+3 , 4k+8≤j≤5k+4 , and 5k+8≤j≤6k+5 .

We also have [figure omitted; refer to PDF] Since the elements succeeding _x6k+6 , _x6k+7 depend on x and y for their values, the cycle begins again with the 6k+6th element, that is, _x0 =_x6k+6 , _x1 =_x6k+7 , ... . Thus, _Pk (...2,3,4...;x,y)=6k+6 and B_Pk (...2,3,4...;x,y)=6k+6 since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by ^x2 .

Secondly, let us consider the 3-generator case. We first note that |x|=4 , |y|=6 , and |z|=8 .

(i) If k=2,_P2 (...2,3,4...;y,z,x)=18 and B_P2 (...2,3,4...;y,z,x)=9 since xθ=^x3 , yθ=^x3 yx , and zθ=x^y2 where θ is an outer automorphism of order 2.

(ii) If k>2,_Pk (...2,3,4...;x,y,z)=6k+6 and B_Pk (...2,3,4...;x,y,z)=6k+6 since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by ^x2 .

The proofs are similar to the proofs of Theorems 3.5.(i) and 3.5.(ii) and are omitted.

Theorem 3.7.

The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the group _A4 are as follows.

If the group is defined by the presentation _A4 =...x,y,z:^x2 =^y3 =^z3 =xyz=e... , then

(i) if k=2 , _P2 (_A4 ;y,z,x)=16 and B_P2 (_A4 ;y,z,x)=4 ,

(ii) if k>2 , [figure omitted; refer to PDF]

where _u1 ,_u2 ,_u3 ∈N , and _hk (3) denote the wall number of the k-step Fibonacci sequence modulo 3.

If the group is defined by the presentation _A4 =...x,y:^x2 =^y3 =^(xy)3 =e... , then

(i[variant prime]): if k=2 , _P2 (_A4 ;x,y)=16 and B_P2 (_A4 ;x,y)=4 ,

(ii[variant prime]): if k>2 , [figure omitted; refer to PDF]

where _u1 ,_u2 ,_u3 ∈N .

Proof.

Firstly, let us consider the 2-generator case. We process as similar to the proof of Theorem 3.6 We first note that |x|=2 , |y|=3 , and |xy|=3 .

(i[variant prime]): If k=2 , we have the sequence for the generating pair ( x , y ),

[figure omitted; refer to PDF] which has period 16 and the basic period 4 since xθ=yx^y2 and yθ=yxy where θ is an outer automorphism of order 4.

(ii[variant prime]): If k>2 ,

let k be even, then the first k elements of sequence for the generating pair ( x , y ) are [figure omitted; refer to PDF] If k≡0mod 4 , [figure omitted; refer to PDF] _Pk (_A4 ;x,y)=3B_Pk (_A4 ;x,y) and B_Pk (_A4 ;x,y)=_u1hk (3) since xθ=yx^y2 and yθ=xyx where θ is an outer automorphism of order 3.

If k≡2mod 4 , [figure omitted; refer to PDF] _Pk (_A4 ;x,y)=2B_Pk (_A4 ;x,y) and B_Pk (_A4 ;x,y)=_u2hk (3) since xθ=x and yθ=xy where θ is a outer automorphism of order 2.

Let k be odd, then the first k elements of sequence are for the generating pair ( x , y ), [figure omitted; refer to PDF] Also, [figure omitted; refer to PDF] _Pk (_A4 ;x,y)=2B_Pk (_A4 ;x,y) and B_Pk (_A4 ;x,y)=_u3hk (3) since xθ=x and yθ=yx where θ is an outer automorphism of order 2.

Secondly, let us consider the 3-generator case. We first note that |x|=2 , |y|=3 , and |z|=3 .

(i) If k=2,_P2 (_A4 ;y,z,x)=16 and B_P2 (_A4 ;y,z,x)=4 since xθ=^y2 xy , yθ=yxy , and zθ=yx where θ is an outer automorphism of order 4.

(ii) If k>2, let k≡0 mod 4 , then _Pk (_A4 ;x,y,z)=3B_Pk (_A4 ;x,y,z) and B_Pk (_A4 ;x,y,z)=_u1hk (3) since xθ=^y2 xy , yθ=xyx , and zθ=zx where θ is an outer automorphism of order 3; let k≡2 mod 4 , then _Pk (_A4 ;x,y,z)=2B_Pk (_A4 ;x,y,z) and B_Pk (_A4 ;x,y,z)=_u2hk (3) since xθ=x , yθ=yx , and zθ=y^z2 where θ is an outer automorphism of order 2; let k be odd; then _Pk (_A4 ;x,y,z)=2B_Pk (_A4 ;x,y,z) and B_Pk (_A4 ;x,y,z)=_u3hk (3) since xθ=x , yθ=xy , and zθ=zx where θ is an outer automorphism of order 2.

The proofs are similar to the proofs of Theorems 3.5.(i) and 3.5.(i.i) and are omitted.

Theorem 3.8.

The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the binary polyhedral group ...2,3,3... are as follows.

If the group is defined by the presentation ...2,3,3...=...x,y,z:^x2 =^y3 =^z3 =xyz... , then

(i) if k=2 , _P2 (...2,3,3...;y,z,x)=48 and B_P2 (...2,3,3...;y,z,x)=12 ,

(ii) if k>2 ,

[figure omitted; refer to PDF] [figure omitted; refer to PDF] where _v1 ,_v2 ∈N , and _hk (6) denote the wall number of the k -step Fibonacci sequence modulo 6.

If the group is defined by the presentation ...2,3,3...=...x,y:^x2 =^y3 =^(xy)3 ... , then

(^{i[variant prime]} ) : if k=2 , _P2 (...2,3,3...;x,y)=48 and B_P2 (...2,3,3...;x,y)=12 ,

(^{ii[variant prime]} ) : if k>2 , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

where _v1 ,_v2 ∈N .

Proof.

Firstly, let us consider the 3-generator case. We first note that |x|=4 , |y|=6 , and |z|=6 .

(i) If k=2,_P2 (...2,3,3...;y,z,x)=48 and B_P2 (...2,3,3...;y,z,x)=12 since xθ=^y2 xy , yθ=x^z4 x , and zθ=^y2 x^y2 where θ is an outer automorphism of order 4.

(ii) If k>2, let k≡0mod 4 , then _Pk (...2,3,3...;x,y,z)=3B_Pk (...2,3,3...;x,y,z) and B_Pk (...2,3,3...;x,y,z)=_v1hk (6) since xθ=yx^y5 , yθ=^z3 xy , and zθ=x^y2 x where θ is an inner automorphism induced by conjugation by ^z3 yx ;

let k[not identical with]0mod 4 , then _Pk (...2,3,3...;x,y,z)=B_Pk (...2,3,3...;x,y,z) and B_Pk (...2,3,3...;x,y,z)=_v2hk (6) since xθ=x , yθ=y , and zθ=z where θ is an inner automorphism induced by conjugation by ^x2 .

The proofs are similar to the proofs of Theorems 3.5.(i) and 3.5.(ii) and are omitted.

Secondly, let us consider the 2-generator case. We first note that |x|=4 , |y|=6 , and |xy|=6 .

(i[variant prime]): If k=2,_P2 (...2,3,3...;x,y)=48 and B_P2 (...2,3,3...;x,y)=12 since xθ=yx^y2 and yθ=^y2 x where θ is an outer automorphism of order 4.

(ii[variant prime]): If k>2 , let k≡0mod 4 , then _Pk (...2,3,3...;x,y)=3B_Pk (...2,3,3...;x,y) and B_Pk (...2,3,3...;x,y)=_v1hk (6) since xθ=^y5 xy , yθ=yx , and zθ=x^y2 x where θ is an inner automorphism induced by conjugation by ^y5 x ,

let k[not identical with]0mod 4 , then _Pk (...2,3,3...;x,y)=B_Pk (...2,3,3...;x,y) and B_Pk (...2,3,3...;x,y)=_v2hk (6) since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by ^x2 .

The proofs are similar to the proofs of Theorem 3.6.(i[variant prime]) and Theorem 3.6.(ii[variant prime]) and are omitted.

Theorem 3.9.

The periods of the k-nacci sequences are k+1 , and the basic period of the basic k-nacci sequences is k+1 in _D2 four-group.

Proof.

We have the presentation _D2 =...x,y:^x2 =^y2 =e, xy=yx... . _Pk (_D2 ;x,y)=k+1 ; see [14] for a proof and B_Pk (_D2 ;x,y)=k+1 since xθ=x and yθ=y where θ is an inner automorphism induced by conjugation by x .

Acknowledgments

The authors thank the referees for their valuable suggestions which improved the presentation of the paper. This paper was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity, Project no. 2010-FEF-61.

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