MATEMATIQKI VESNIK

58 (2006), 47–56

UDK 511.311

originalni nauqni rad research paper

ON THE PERIODS OF 2-STEP GENERAL FIBONACCI SEQUENCES IN DIHEDRAL GROUPS

Erdal Karaduman and Hüseyin Aydin

Abstract. In this paper, we investigate the simply periodic cases of 2-step general Fibonacci sequences in dihedral groups D_{n}, and we also find the period of the sequences if the sequences are simply periodic.

1. Introduction

The study of Fibonacci sequences in groups began with the earlier work of Wall [9] where the ordinary Fibonacci sequences in cyclic groups were investigated. In the mid-eighties, Wilcox extended the problem to abelian groups [10]. Prolific co-operation of Campbell, Doostie and Robertson expanded the theory to some finite simple groups [3]. Aydin and Smith proved in [2] that the lengths of ordinary 2-step Fibonacci sequences are equal to the lengths of ordinary 2-step Fibonacci recurrences in finite nilpotent groups of nilpotency class 4 and a prime exponent. The theory has been generalized in [4,5] to the ordinary 3-step Fibonacci sequences in finite nilpotent groups. Then, it is shown in [1] that the period of 2-step general Fibonacci sequence is equal to the length of fundamental period of the 2-step general recurrence constructed by two generating elements of the group of exponent p and nilpotency class 2. Karaduman and Yavuz showed that the periods of the 2-step Fibonacci recurrences in finite nilpotent groups of nilpotency class 5 and a prime exponent are p · k(p), for 2 < p ⩽ 2927, where p is prime and k(p) are the periods of ordinary 2-step Fibonacci sequences [7]. The theory has been generalized in [6] to the 2-step general Fibonacci sequences in finite nilpotent groups of nilpotency class 4 and exponent p. Knox proved that periods of k-nacci (k-step Fibonacci) sequences in dihedral group were equal to 2k + 2 [8].

This paper is related to period of 2-step general Fibonacci sequences in dihedral groups. It not only addresses to the question whether the sequences are simply periodic or not, but it also gives an upper bound for period of the sequences when n is odd.

AMS Subject Classification: 11 B 39

Keywords and phrases: Fibonacci sequences, period, dihedral group.

A k-nacci sequence in a finite group is a sequence of group elements x_{0}, x_{1}, x_{2}, …, x_{n}, … for which, given an initial (seed) set x_{0}, x_{1}, x_{2}, …, x_{j−1}, each element is defined by

x_{n} = \{ \begin{matrix}x_{0}x_{1}x_{2} ⋯ x_{n−1}, & for j ⩽ n < k, \\ x_{n−k}x_{n−k+1} ⋯ x_{n−1}, & for n ⩾ k. \end{matrix}

We also require that the initial elements of the sequence, x_{0}, x_{1}, x_{2}, …, x_{j−1}, generate a group, thus forcing the k-nacci sequence to reflect the structure of the group. The k-nacci sequence of a group generated by x_{0}, x_{1}, x_{2}, … x_{j−1} is denoted by f_{k}(G; x_{0}, x_{1}, …, x_{j−1}).

2-step Fibonacci sequence in the integers modulo m can be written as F_{2}(Z_{m}; 0, 1). We call a 2-step Fibonacci sequence of a group elements a Fibonacci sequence of a finite group. A finite group G is k-nacci sequenceable if there exists a k-nacci sequence of G such that every element of the group appears in the sequence.

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 period of the sequence. For example, the sequence a, 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, … is simply periodic with period 6.

A group D_{n} is dihedral if

D_{n} = ⟨a, b : a^{n} = e, b^{2} = e, ab = ba^{−1}⟩.

The order of the group D_{n} is 2n. Note that in a dihedral group generated by a and b,

(ab)^{2} = abab = aa^{−1}bb = e.

Theorem 1.1. [8] A k-nacci sequence in a finite group G is simply periodic.

Theorem 1.2. [8] For some n ⩾ 3, P_{k}(D_{n}; a, b) = 2k + 2.

2.Main results and proof

Firstly, we give a lemma which will be used in proofs of our theorems.

Lemma 2.1. In the dihedral group

D_{n} = ⟨a, b : a^{n} = e, b^{2} = e, ab = ba^{−1}⟩,

(i) (a^{x}b)^{y} = e, for x = 1, 2, …, n − 1, where y is any even integer. (ii) (a^{x}b)^{z} = (a^{x}b), for x = 1, 2, …, n − 1, where z is any odd integer.

Proof. (i) Let y be an even integer. We prove this fact by induction argument on y. If y = 2, then we have

(a^{x}b)^{2} = a^{x}ba^{x}b = a^{x}a^{−1}ba^{x−1}b = a^{x}a^{−2}ba^{x−2}b = ⋯ = a^{x}a^{−x}bb = e.

So, our argument holds for y = 2. Now, we assume that

(a^{x}b)^{2t} = e.

Multiplying both sides by (a^{x}b)^{2}, we have

(a^{x}b)^{2t}(a^{x}b)^{2} = (a^{x}b)^{2t+2} = e(a^{x}b)^{2} = ee = e.

Hence, we are done.

(ii) Let z = 2t + 1 be an odd integer. By (i), we have (a^{x}b)^{2t} = e. Thus,

(a^{x}b)^{2t+1} = (a^{x}b)^{2t}(a^{x}b) = e(a^{x}b) = (a^{x}b).

So, we are done.

Let n ⩾ 3. Consider the dihedral group D_{n} with generators a, b. The 2-step general Fibonacci sequences in D_{n} are defined as

x_{0} = a, x_{1} = b and for i ⩾ 2, x_{i} = x_{i−2}^{m}x_{i−1}^{l},

where m, l are integers.

Theorem 2.2. In D_{n}, we have

(i) If m ≡ 0 (mod n), then 2-step general Fibonacci sequences in D_{n} are not simply periodic for any l.

(ii) If m is any even integer, then 2-step general Fibonacci sequences in D_{n} are not simply periodic for any l.

(iii) If m | n such that m ≠ 1, then 2-step general Fibonacci sequences in D_{n} are not simply periodic for any l.

Proof. (i) Let m ≡ 0 (mod n). Then, there exists an x ∈ Z such that m = nx. Thus, the sequence is

x_{0} = a, x_{1} = b, x_{2} = b^{l}, x_{3} = b^{m}b^{l^{2}}, …

Since, the elements succeeding x_{2}, x_{3} do not depend on a, b and the cycle does not begin again with a, b, the sequence is not simply periodic. We are done.

(ii) Let m be an even integer such that m = 2t, t ∈ Z. If l is an even integer, then the sequence will be

x_{0} = a, x_{1} = b, x_{2} = a^{2t}, x_{3} = a^{2tl}, …

Since b^{l} = e, the elements succeeding x_{2}, x_{3} does not include b and the cycle does not begin again with a, b, the sequence is not simply periodic. If l is an odd integer, then the sequence will be

x_{0} = a, x_{1} = b, x_{2} = a^{2t}b, x_{3} = a^{2t}b, …

Since (a^{2t}b)^{m} = e, b^{l} = b and (a^{2t}b)^{l} = a^{2t}b, from Lemma 1,(i) and (ii), the cycle does not begin again with a, b, so the sequence is not simply periodic. We are done.

(iii) Let m | n such that m ≠ 1. Then, there exists an x ∈ Z such that n = mx. Notice that m is odd. If x = 1, then m = n and the sequence will not be simply periodic, from Theorem 2.2.(i). Therefore, two cases occur, which are: l is odd and l is even.

Case 1. Let l be an odd integer. We know that (a^{m}b)^{l} = a^{m}b and b^{l} = b, from Lemma 1,(ii). Hence, the sequence will be

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{n−m}, …

Notice that we have n − m = m or n − m = mt (t ∈ Z).

I. If n − m = m, then the sequence will be

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{m}, x_{4} = a^{m}ba^{ml}, …

Notice that, the cases

(1) ml ≡ 0 (mod n), (2) ml ≡ mt (mod n)

occur. Also, in the subcases (1) and (2), the cases mm ≡ 0 (mod n) and mm ≡ m (mod n) occur. Similarly,

II. If n − m = mt (t ∈ Z), then the cases

(1′) (mt)l ≡ 0 (mod n), (2′)(mt)l ≡ mt (mod n), t ∈ Z

occur. Also, in the cases (1’) and (2’), the subcases, (mt)m ≡ 0 (mod n) and (mt)m ≡ mt (mod n) occur.

Now, for n − m = m, if ml ≡ 0 (mod n), then the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{m}, x_{4} = a^{m}b, x_{5} = a^{m}b, x_{6} = e, …

(mm ≡ 0 (mod n)) or

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{m}, x_{4} = a^{m}b, x_{5} = a^{2m}b, x_{6} = a^{m}, x_{7} = a^{m}b,

… (mm ≡ m (mod n)). Thus, it can be easily seen that the cycle does not begin again with a, b in any of the cases. Therefore, the sequence is not simply periodic.

Now, for n − m = m, if ml ≡ m (mod n), then the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{m}, x_{4} = b, x_{5} = b, x_{6} = e, …

(mm ≡ 0 (mod n)) or

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{m}, x_{4} = b, x_{5} = a^{m}b, x_{6} = a^{m},

… (mm ≡ m (mod n)). Thus, it can be easily seen that the cycle does not begin again with a, b, in both cases. Therefore, the sequence is not simply periodic.

Similarly, we study case II.

Now, for n − m = mt (t ∈ Z), if (mt)l ≡ 0 (mod n), then the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{mt}, x_{4} = a^{m}b,
x_{5} = a^{m}b, x_{6} = e, x_{7} = a^{m}b, x_{8} = a^{m}b, x_{9} = e, … ((mt)m ≡ 0 (mod n))

or

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{mt}, x_{4} = a^{m}b, x_{5} = a^{mt+m}b,
x_{6} = a^{−mt}, x_{7} = a^{mt+m}b, x_{8} = a^{mt+m}b, x_{9} = e, … ((mt)m ≡ mt (mod n)).

Thus, it can be easily seen that the cycle does not begin again with a, b in both cases. Therefore, the sequence is not simply periodic.

Now, for n − m = mt, if (mt)l ≡ mt (mod n), then the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{mt}, x_{4} = a^{m−mt}b, x_{5} = a^{m−mt}b, x_{6} = e, …

((mt)m ≡ 0 (mod n)) or

x_{0} = a, x_{1} = b, x_{2} = a^{m}b, x_{3} = a^{mt}, x_{4} = a^{m−mt}b,
x_{5} = a^{m}b, x_{6} = a^{−mt}, x_{7} = a^{mt+m}b, … ((mt)m ≡ mt (mod n)).

Thus, it can be easily seen that the cycle does not begin again with a, b in both cases. Therefore, the sequence is not simply periodic.

Case 2. Let l be an even integer. We know that (a^{m}b)^{l} = e and b^{l} = e, from Lemma 1,(i). Hence, the sequence will be

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, …

In this case, two subcases, which are n − ml ≡ 0 (mod n) and n − ml ≡ mt (mod n) occur. In other words, we have n − ml = 0 or n − ml = mt (t ∈ Z). For each of these cases, the possible subcases are mm ≡ 0 (mod n), mm ≡ m (mod n) and mm ≡ mv (mod n).

If ml ≡ 0 (mod n), then the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = b, x_{4} = e, … for mm ≡ 0 (mod n),
x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = b, x_{4} = a^{m}, … for mm ≡ m (mod n),
x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = b, x_{4} = a^{m^{2}}, x_{5} = a^{n−m^{2}l}, x_{6} = a^{m^{3}},
… for mm ≡ mv (mod n), (v ∈ Z).

For the first two subcases, it is clear that the sequence is not simply periodic. Notice that, since m | n, after certain steps m^{v} ≡ m (mod n) and (n − m^{2}l − m^{3}l − ⋯ − m^{u}l) ≡ 0 (mod n) in the third subcase. Thus, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = b, x_{4} = a^{m^{2}}, x_{5} = a^{n−m^{2}l},
x_{6} = a^{m^{3}}, … x_{η} = b, x_{η+1} = a^{m}, …

which is not simply periodic.

If ml ≡ mt (mod n), the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, x_{4} = e, x_{5} = a^{n−ml}b, … for mm ≡ 0 (mod n), which is not simply periodic. On the other hand, when ml ≡ mt (mod n) if mm ≡ m (mod n), we have,

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, x_{4} = a^{m}, x_{5} = a^{n−2ml}b, x_{6} = a^{m}, …

for mm ≡ m (mod n). Since m | n, after certain steps, we have (n − m^{2}l − m^{3}l − ⋯ − m^{u}l) ≡ 0 (mod n). Thus, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, x_{4} = a^{m},
x_{5} = a^{n−2ml}b, …, x_{μ} = b, x_{μ+1} = a^{m}, … for mm ≡ m (mod n),

which is not simply periodic.

Now, when ml ≡ mt (mod n), if mm ≡ mv (mod n) the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, x_{4} = a^{m^{2}},
x_{5} = a^{n−ml−m^{2}l}b, x_{6} = a^{m^{3}}, …

Since m | n, in third subcase, after certain steps there are v, u ∈ Z such that m^{v} ≡ m (mod n) and (n − ml − m^{2}l − ⋯ − m^{u}l) ≡ 0 (mod n). Thus, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{m}, x_{3} = a^{n−ml}b, x_{4} = a^{m^{2}}, x_{5} = a^{n−ml−m^{2}l}b,
x_{6} = a^{m^{3}}, … x_{ζ} = b, x_{ζ+1} = a^{m}, …

which is not simply periodic.

Hence, we can say that the Theorem 1.1 does not hold for 2-step general Fibonacci sequences in dihedral groups.

Corollary. If m ≡ d (mod n) such that d | n, then 2-step general Fibonacci sequences in D_{n} are not simply periodic.

Theorem 2.3. Let n ⩾ 3. Consider the dihedral group D_{n} with generators a, b.

(i) If m ≡ 1 (mod n) and l = [n, 2]t, t ∈ Z, then 2-step general Fibonacci sequences in D_{n} are simply periodic with period 2, where [n, 2] denotes the least common multiple of n and 2.

(ii) If m ≡ (n − 1) (mod n) and l ≡ (n − 1) (mod n) such that l is an odd integer, then 2-step general Fibonacci sequences in D_{n} are simply periodic with period 3.

(iii) If m ≡ 1 (mod n) and l ≡ 1 (mod n) such that l is an odd integer, then 2-step general Fibonacci sequences in D_{n} are simply periodic with period 6.

Proof. (i) Let m ≡ 1 (mod n) and l = [n, 2]t, t ∈ Z. Then, there exists an x ∈ Z such that m = nx+1. Notice that m is odd and l is even. Thus, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a, x_{3} = b, …

Since the elements succeeding x_{2}, x_{3} depend on a and b for their values, the cycle begins again with the 2nd element. Thus, the 2-step general Fibonacci sequences in D_{n} are simply periodic with period 2.

(ii) Let m ≡ (n − 1) (mod n) and l ≡ (n − 1) (mod n) such that l is an odd integer. Notice that m, l are odd. Thus, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a^{n−1}b, x_{3} = ba^{n−1}b = a, x_{4} = a^{n−1}ba^{n−1} = b, …

Since the elements succeeding x_{3}, x_{4} depend on a and b for their values, the cycle begins again with the 3rd element. Hence, the 2-step general Fibonacci sequences in D_{n} are simply periodic with period 3.

Notice that the conditions m ≡ (n − 1) (mod n), l ≡ (n − 1) (mod n) and l – odd, must hold at the same time. Otherwise, the period of the sequence may not be equal to 3. For example, if m = 13 ≡ 6 (mod 7) and l = 6 ≡ 6 (mod 7), the period of the 2-step general Fibonacci sequences in D_{7} is 4.

(iii) Let m ≡ 1 (mod n) and l ≡ 1 (mod n) such that l is an odd integer. The sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = ab, x_{3} = a^{−1} = a^{n−1}, x_{4} = ab(a^{l})^{−1} = a^{2}b,
x_{5} = (a^{m})^{−1}a^{2}b = ab, a_{6} = a, x_{7} = b, …

Since the elements succeeding x_{6}, x_{7} depend on a and b for their values, the cycle begins again with the 6th element. Thus, the 2-step general Fibonacci sequences in D_{n} are simply periodic with period 6.

Theorem 2.4. Let n be an odd integer and n ⩾ 3. Consider the dihedral group D_{n} with generators a, b.

(i) If m ≡ 1 (mod n) and l is even such that l ≠ [n, 2]t, t ∈ Z, then 2-step general Fibonacci sequences in D_{n} are simply periodic with period 2n, where [n, 2] denotes the least common multiple of n and 2.

(ii) If m ≡ (n − 1) (mod n) and l ≢ (n − 1) (mod n) such that l is an odd integer, then 2-step general Fibonacci sequences in D_{n} are simply periodic with period 3n.

Proof. (i) Let m ≡ 1 (mod n) and l is even such that l ≠ [n, 2]t, t ∈ Z. Then, the sequence reduces to

x_{0} = a, x_{1} = b, x_{2} = a, x_{3} = a^{n−l}b, x_{4} = a, x_{5} = a^{n−2l}b, …,
x_{n−2} = a^{n−( \frac{n−1}{2} −1)l}b, x_{n−1} = a, x_{n} = a^{n− \frac{n−1}{2} l}b, …, x_{2n−2} = a,
x_{2n−1} = a^{n−(n−1)l}b, x_{2n} = a, x_{2n+1} = b, …

Since the elements succeeding x_{2n}, x_{2n=1} depend on a and b for their values, the cycle begins again with the 2n-th element. Thus, 2-step general Fibonacci sequences in D_{n} are simply periodic with period 2n.

(ii) Let m ≡ (m − 1) (mod n) and l ≢ (n − 1) (mod n) such that l is an odd integer. Then, l ≡ 0, 1, … (n − 2) (mod n). So, 2-step general Fibonacci sequences in D_{n} reduce to

x_{0} = a, x_{1} = b, x_{2} = a^{n−1}b, x_{3} = a, x_{4} = a^{n−1−l}b, x_{5} = a^{2n−2−l}b,
x_{6} = a, …, x_{3n−1} = a^{n−n−(n−1)l}b, x_{3n} = a, x_{3n+1} = b, …

Since the elements succeeding x_{3n}, x_{3n+1} depend on a and b for their values, the cycle begins again with the 3n-th element. 2-step general Fibonacci sequences in D_{n} are simply periodic with period 3n.

The periods of 2-step general Fibonacci sequences in some D_{n} are given in tables 1, 2.

Period  D_{3}  D_{4}  D_{5}
2  m ≡ 1 (mod 3), l = [3, 2]t  m ≡ 1 (mod 4), l = [4, 2]t  m ≡ 1 (mod 5), l = [5, 2]t
3  m ≡ 2 (mod 3),  m ≡ 3 (mod 4),  m ≡ 4 (mod 5),
l ≡ 2 (mod 3)  l ≡ 3 (mod 4)  l ≡ 4 (mod 5)
where m, l are odd  where m, l are odd  where m, l are odd
4  m ≡ 2 (mod 3), l is even  1. m ≡ 3 (mod 4), l = [4, 2]t  m ≡ 4 (mod 5), l is even

2. m is odd, l is even, but l ≠ [4, 2]t

6 m ≡ 1 (mod 3), l ≠ [3, 2]t 1. m is odd, l ≡ 1 (mod 4) m ≡ 1 (mod 5), l is odd
2. m ≡ 1 (mod 4), l ≡ 1 (mod 4)
8  m ≡ 2, 3 (mod 5), l is even
9  m ≡ 2 (mod 3),

l ≢ 2 (mod 3), l is odd

10  m ≡ 1 (mod 5)

l ≠ [5, 2]t, l is even

12  m ≡ 2, 3 (mod 5), l is odd
15  m ≡ 4 (mod 5),

l ≢ 4 (mod 5), l is odd

Table 1.

Period  D_{6}  D_{7}  D_{8}
2  m ≡ 1 (mod 6), l = [6, 2]t m ≡ 3 (mod 7), l = [7, 2]t m ≡ 1 (mod 8), l = [8, 2]t
3  m ≡ 5 (mod 6),  m ≡ 6 (mod 7),  m ≡ 7 (mod 8),
l ≡ 5 (mod 6)  l ≡ 6 (mod 7)  l ≡ 7 (mod 8)
where m, l are odd  where m, l are odd  where m, l are odd
4  m ≡ 5 (mod 6), l is even m ≡ 6 (mod 7), l is even  1. m ≡ 3, 7 (mod 8),
l ≡ 2, 6 (mod 8)
2. m ≡ 1, 3, 5, 7 (mod 8)
l ≡ 4 (mod 8)
3. m ≡ 3, 5, 7 (mod 8)
l ≡ 0 (mod 8)
6  m ≡ 1 (mod 6), l ≠ [6, 2]t 1. m ≡ 1 (mod 7), l is odd  1. m ≡ 1, 5 (mod 8),
l ≡ 1, 5 (mod 8)
2. m ≡ 2 (mod 7), l is even 2. m ≡ 1, 3, 5, 7 (mod 8)
l ≡ 3 (mod 8)
3. m ≡ 1, 3, 5 (mod 8),
l ≡ 7 (mod 8)
8  m ≡ 1, 5 (mod 8),
l ≡ 2, 6 (mod 8)
9  m ≡ 5 (mod 6),  m ≡ 3, 5 (mod 7),
l ≡ 1, 3 (mod 6)  l is odd

10

12  m ≡ 3, 5 (mod 7),  m ≡ 3, 5 (mod 8)
l is even  l ≡ 1, 5 (mod 8)
14  m ≡ 1 (mod 7)

l ≠ [7, 2]t, l is even

18  m ≡ 2, 4 (mod 7), l is odd
21  m ≡ 6 (mod 7),

l ≢ 6 (mod 7), l is odd

Table 2.

REFERENCES

[1] Aydın, H., Dikici, R., General Fibonacci sequences in finite groups, Fibonacci Quarterly 36, 3 (1998), 216–221.

[2] Aydın, H., Smith, G. C., Finite p-quotients of some cyclically presented groups, J. London Math. Soc. 49, 2 (1994), 83–92.

[3] Campbell, C. M., Doostie, H., Robertson, E. F., Fibonacci Length of Generating Pairs in Groups, Applications of Fibonacci Numbers 3, 27–35. Ed. G. E. Bergum et al., Kluwer Academic Publishers, 1990.

[4] Dikici, R., Smith, G. C., Recurrences in finite groups, Turkish J. Math. 19 (1995), 321–329.

[5] Dikici, R., Smith, G. C., Fibonacci sequences in finite nilpotent groups, Turkish J. Math. 21 (1997), 133–142.

[6] Karaduman, E., Aydın, H., General 2-step Fibonacci sequences in nilpotent groups of exponent p and nilpotency class 4, Appl. Math. Comput. 141 (2003), 491–497.

[7] Karaduman, E., Yavuz, U., On the period of Fibonacci sequences in nilpotent groups, Appl. Math. Comput. 142 (2003), 321–332.

[8] Knox, S. W., Fibonacci sequences in finite groups, Fibonacci Quarterly 30, 2 (1992), 116–120.

[9] Wall, D. D., Fibonacci series module, Amer. Math. Monthly 67 (1969), 525–532.

[10] Wilcox, H. J., Fibonacci sequences of period n in groups, Fibonacci Quarterly 24 (1986), 356–361.

(received 03.09.2004, in revised form 26.10.2005)

Ataturk Üniversitesi Fen Edebiyat Fakültesi Matematik Bölümü, 25240 Erzurum/TURKEY

E-mail: eduman@atauni.edu.tr