Fibonacci lengths of all finite $p$-groups of exponent $p^2$
The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent $p$ and the lengths depend on the Wall number $k(p)$. The p-groups of nilpotency class 3 and exponent $p$ were studied in 2004 also by Dikici. In the paper, we study all $p...
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| author | Ahmadi, B. Doostie, H. Ахмаді, Б. Доостіе, Г. |
| author_facet | Ahmadi, B. Doostie, H. Ахмаді, Б. Доостіе, Г. |
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| datestamp_date | 2020-03-18T19:15:53Z |
| description | The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent $p$ and the lengths depend on the Wall number $k(p)$. The p-groups of nilpotency class 3 and exponent $p$ were studied in 2004 also by Dikici. In the paper, we study all $p$-groups of nilpotency class 3 and exponent $p^2$. Thus, we complete the study of Fibonacci lengths of all $p$-groups of order $p^4$ by proving that the Fibonacci length is $k(p^2)$. |
| first_indexed | 2026-03-24T02:23:34Z |
| format | Article |
| fulltext |
UDC 512.5
B. Ahmadi (Islamic Azad Univ., Lahijan, Iran),
H. Doostie (Islamic Azad Univ., Tehran, Iran)
FIBONACCI LENGTHS OF ALL FINITE p-GROUPS OF EXPONENT p2
ДОВЖИНИ ФIБОНАЧЧI ДЛЯ ВСIХ СКIНЧЕННИХ p-ГРУП ЕКСПОНЕНТИ p2
The Fibonacci lengths of finite p-groups were studied by Dikici and co-authors since 1992. All of the considered groups
are of exponent p, and the lengths depend on the Wall number k(p). The p-groups of nilpotency class 3 and exponent p
were studied in 2004 also by Dikici. In the present paper, we study all p-groups of nilpotency class 3 and exponent p2. We
thus complete the study of Fibonacci lengths of all p-groups of order p4, proving that the Fibonacci length is k(p2).
Довжини Фiбоначчi скiнченних p-груп вивчалися Дiкiчi та спiвавторами з 1992 року. Всi групи, що розглядалися,
були групами експоненти p, а всi довжини залежали вiд числа Уолла k(p). p-Групи класу нiльпотентностi 3 i
експоненти p були також дослiдженi Дiкiчi у 2004 роцi. У данiй статтi ми вивчаємо всi p-групи класу нiльпотентностi
3 i експоненти p2. Цим завершується дослiдження довжини Фiбоначчi всiх p-груп порядку p4; при цьому доведено,
що довжина Фiбоначчi дорiвнює k(p2).
1. Introduction. The study of Fibonacci sequences in groups began with the earlier work of Wall
[19] in 1960, where the ordinary Fibonacci sequences in cyclic groups were investigated. In the
mid-eighties, Wilcox [20] extended the problem to the abelian groups. In 1990, Campbell et al. [5]
expanded the theory to some classes of finite groups. In 1992, Knox proved that the periods of k-
nacci (k-step Fibonacci) sequences in the dihedral groups are equal to 2k + 2, in the article [17]. In
the progress of this study, the article [2] of Aydin and Smith proves that the lengths of the ordinary
2-step Fibonacci sequences are equal to the lengths of the ordinary 2-step Fibonacci recurrences in
finite nilpotent groups of nilpotency class 4 and a prime exponent, in 1994.
Since 1994, the theory has been generalized and many authors had nice contributions in compu-
tations of recurrence sequences in groups and we may give here a brief of these attempts. In [7] and
[8] the definition of the Fibonacci sequence has been generalized to the ordinary 3-step Fibonacci
sequences in finite nilpotent groups. Then in the article [1] it is proved that the period of 2-step
general Fibonacci sequence is equal to the length of the fundamental period of the 2-step general
recurrence constructed by two generating elements of a group of nilpotency class 2 and exponent p.
In [16] 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 a prime and k(p) is the period of ordinary 2-step Fibonacci sequence. The main role of the articles
[14] and [15] in generalizing the theory was to study the 2-step general Fibonacci sequences in finite
nilpotent groups of nilpotency class 4 and exponent p and to the 2-step Fibonacci sequences in finite
nilpotent groups of nilpotency class n and exponent p, respectively.
Going on a detailed literature in this area of research, we have to mention certain essential com-
putation on the Fibonacci lengths of new structures like the semidirect products, the direct products
and the automorphism groups of finite groups which have been studied in the articles [3, 4, 9 – 12].
c© B. AHMADI, H. DOOSTIE, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5 603
604 B. AHMADI, H. DOOSTIE
Let s = (si) be the 2-step Fibonacci sequence of numbers defined by s0 = 0, s1 = 1, si =
= si−2 + si−1, for i ≥ 2. We may extend the sequence backwards to obtain a bi-infinite sequence.
The fundamental period or Wall number (see [19]) of this sequence is denoted by k(s, pn), where
the sequence reduced modulo pn, for a positive integer n and a prime p. Since now on, we denote
k(s, pn) by k(pn).
A 2-step general Fibonacci sequence in a finite non-abelian 2-generated group G = 〈a, b〉 is
defined by x0 = a, x1 = b, xi = xmi−2x
l
i−1, for i ≥ 2 and the integers m and l. If m = l = 1, the
least period of this sequence is called the Fibonacci length of G and denoted by k(G).
Among all of the p-groups of order p4 and nilpotency class 3 (see [18]), the group
H = 〈a, b, c, d | ap = bp = cp = dp = 1, [a, b] = [a, c] = [a, d] = 1,
[b, d] = a, [c, d] = b〉, p 6= 3,
is of exponent p and studied by Dikici [6]. The remained four classes indeed, the groups
K = 〈a, b, c | a9 = b3 = c3 = 1, [a, b] = 1, [a, c] = b, [c, b−1] = a−3〉,
and
Lα = 〈a, b, c | ap2 = bp = 1, cp = aαp, [a, b] = ap, [a, c] = b, [b, c] = 1〉,
where α = 0, 1, or a non-residue modulo p, are of exponent p2. The aim of this paper is to study the
Fibonacci lengths of these groups. First of all we attempt to give a power-commutator presentation
for the groups (see [13]) and by investigating their nilpotency class we will go to the computation of
Fibonacci lengths.
Our main result is:
Main theorem. For a group G of order p4 and of exponent p2 which is of nilpotency class 3,
k(G) = k(p2) where, p is an odd prime.
The proof of this theorem and the computation of k(G) for the group G = K, may be checked
by using a procedure in a group theoretic software like GAP (GAP-groups, Algorithms and Pro-
gramming, Ver. gap4r4p12; http://www.gap-system.org). Of course, we will give the details of our
calculation on k(G) of the group G = Lα in the next section. Also, we will state a conjecture for the
groups of orders of p5, p6 and p7.
2. The groups Lα. Case α = 0. Let G = Lα, where α = 0. Then G = 〈a, b, c | ap2 = bp =
= cp = 1, [a, b] = ap, [a, c] = b, [b, c] = 1〉. By the relations of group, ap ∈ [G,G′]. Therefore, G
has nilpotency class 3 and [G,G′] ≤ Z(G). Hence ap is a central element of G. A power-commutator
presentation of G may be given as follows:
G =
〈
x, y, z, w | xp = yp = zp = 1, wp = x, [x, y] = [x, z] = [x,w] = 1,
[z, y] = 1, [w, y] = x, [w, z] = y
〉
.
Case α = 1. Let G = Lα, where α = 1. Then G = 〈a, b, c | ap2 = bp = 1, cp = ap, [a, b] =
= ap, [a, c] = b, [b, c] = 1〉. We may show that G has the following power-commutator presentation:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
FIBONACCI LENGTHS OF ALL FINITE p-GROUPS OF EXPONENT p2 605
G =
〈
x, y, z, w | xp = yp = 1, zp = wp = x, [x, y] = [x, z] = [x,w] = 1,
[z, y] = 1, [w, y] = x, [w, z] = y
〉
.
Case where α is a non-residue modulo p. Let G = Lα, where α is a non-residue modulo p.
Then G = 〈a, b, c | ap2 = bp = 1, cp = aαp, [a, b] = ap, [a, c] = b, [b, c] = 1〉. We may show that
G has the following power-commutator presentation:
G =
〈
x, y, z, w | xp = yp = 1, zp = xα, wp = x, [x, y] = [x, z] = [x,w] = 1,
[z, y] = 1, [w, y] = x, [w, z] = y
〉
.
Note that in the new presentations, the group G is generated by w and z. Moreover, x is a central
element. Also, each element of G can be uniquely represented as xaybzcwd, where in the first case
a, b, c reduced modulo p and d reduced modulo p2 and in the second and third cases a and b reduced
modulo p and c and d reduced modulo p2. From now on we suppose that G = Lα, where α = 0, 1,
or a non-residue modulo p. First we prove some elementary results.
Lemma 2.1. For every positive integers m and n,
(i) wmyn = xmnynwm,
(ii) wmzn = x(
m+1
2 )nymnznwm.
Proof. Since x is a central element of G, then (i) may be proved by the induction method. To
prove (ii) we may use (i) and the relation [z, y] = 1.
Lemma 2.2. Let xaybzcwd and xa
′
yb
′
zc
′
wd
′
be elements of G. Then
(xaybzcwd)(xa
′
yb
′
zc
′
wd
′
) = xa+a
′+db′+(d+1
2 )c′yb+b
′+dc′zc+c
′
wd+d
′
.
Proof. By using Lemma 2.1, we have
(xaybzcwd)(xa
′
yb
′
zc
′
wd
′
) = xa+a
′
ybzcwdyb
′
zc
′
wd
′
=
= xa+a
′
ybzcxdb
′
yb
′
wdzc
′
wd
′
= xa+a
′+db′yb+b
′
zcwdzc
′
wd
′
=
= xa+a
′+db′yb+b
′
zcx(
d+1
2 )c′ydc
′
zc
′
wdwd
′
= xa+a
′+db′+(d+1
2 )c′yb+b
′+dc′zc+c
′
wd+d
′
.
Lemma 2.3. Let xaybzcwd and xa
′
yb
′
zc
′
wd
′
be elements of G and m and l be positive integers.
Then
(i) (xaybzcwd)m = xma+(
m
2 )bd+(
m
2 )c(
d+1
2 )+(m3 )cd
2
ymb+(
m
2 )cdzmcwmd,
(ii) (xaybzcwd)m(xa
′
yb
′
zc
′
wd
′
)l = xa
′′
yb
′′
zc
′′
wd
′′
,
where
a′′ = ma+
(
m
2
)
bd+
(
m
2
)
c
(
d+ 1
2
)
+
(
m
3
)
cd2+
+la′ +
(
l
2
)
b′d′ +
(
l
2
)
c′
(
d′ + 1
2
)
+
(
l
3
)
c′d′2+
+mldb′ +m
(
l
2
)
dc′d′ +
(
md+ 1
2
)
lc′,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
606 B. AHMADI, H. DOOSTIE
b′′ = mb+
(
m
2
)
cd+ lb′ +
(
l
2
)
c′d′ +mldc′,
c′′ = mc+ lc′,
d′′ = md+ ld′.
Proof. (i) By induction on m. (ii) By using (i) and Lemma 2.2.
Lemma 2.4. Every element of the Fibonacci sequence in the group G may be presented by
tn = xanybnzsnwsn−1 , where the sequences {an}∞0 and {bn}∞0 are defined as follows:
b0 = 0, bn =
n−1∑
i=0
sn−1−isi−1si+1, n ≥ 1,
a0 = 0, an =
n−1∑
i=0
sn−1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
, n ≥ 1.
Proof. We use an induction method on n. It is obvious that t0 = w = xa0yb0zs0ws−1 and
t1 = z = xa1yb1zs1ws0 , for, a1 = b1 = 0. Now assume that the result holds for n and n+ 1, where
n ≥ 0. Then
tn+2 = tntn+1 = (xanybnzsnwsn−1)(xan+1ybn+1zsn+1wsn) =
= xan+an+1+sn−1bn+1+(sn−1+1
2 )sn+1ybn+bn+1+sn−1sn+1zsn+sn+1wsn−1+sn =
= xa
′
yb
′
zsn+2wsn+1 ,
where
a′ = an + an+1 + sn−1bn+1 +
(
sn−1 + 1
2
)
sn+1 =
=
n−1∑
i=0
sn−1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
+
+
n∑
i=0
sn−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
+ sn−1bn+1 +
(
sn−1 + 1
2
)
sn+1 =
=
n∑
i=0
sn−1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
− s−1
(
sn−1bn+1 +
(
sn−1 + 1
2
)
sn+1
)
+
+
n∑
i=0
sn−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
+ sn−1bn+1 +
(
sn−1 + 1
2
)
sn+1 =
=
n∑
i=0
sn+1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
=
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
FIBONACCI LENGTHS OF ALL FINITE p-GROUPS OF EXPONENT p2 607
=
n+1∑
i=0
sn+1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
= an+2
and
b′ = bn + bn+1 + sn−1sn+1 =
=
n−1∑
i=0
sn−1−isi−1si+1 +
n∑
i=0
sn−isi−1si+1 + sn−1sn+1 =
=
n∑
i=0
sn−1−isi−1si+1 − s−1sn−1sn+1 +
n∑
i=0
sn−isi−1si+1 + sn−1sn+1 =
=
n∑
i=0
sn+1−isi−1si+1 = bn+2.
Lemma 2.4 is proved.
From now on we shall be working modulo p2. Let k = k(p2). The following equations hold and
are easy to see:
sk−i = s−i = (−1)i+1si,
k−1∑
i=0
si =
k−1∑
i=0
sk−i,
k−1∑
i=0
si+a =
k−1∑
i=0
si, a ∈ Z.
The proofs of the Lemmas 2.5, 2.6 and 2.7 may be found in [2] and [6].
Lemma 2.5. The following equations hold:
(i)
∑k−1
i=0
si = 0,
(ii)
∑k−1
i=0
s2i = 0,
(iii)
∑k−1
i=0
s3i = 0.
Lemma 2.6. If p > 3, then
(i)
∑k−1
i=0
sisi−1 = 0,
(ii)
∑k−1
i=0
s2i−1si =
∑k−1
i=0
si−1s
2
i = 0.
Lemma 2.7. For every integers a, b, c, d, and e the following equations hold:
(i)
∑k−1
i=0
si+asi+bs−i+csi = 0,
(ii)
k−1∑
i=0
∑i−1
j=0
s−i+asi+bsi−j−dsj+esi+c = 0.
Lemma 2.8. The following equations hold:
(i)
∑k−1
i=0
(−1)is3i = 0,
(ii)
∑k−1
i=0
(−1)is2i−1si =
∑k−1
i=0
(−1)isi−1s2i = 0, p > 3.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
608 B. AHMADI, H. DOOSTIE
Proof. (i)
k−1∑
i=0
(−1)is3i−1 =
k−1∑
i=0
s3−(i−1) =
k−1∑
i=0
s3k−(i−1) =
k−1∑
i=0
s3i = 0.
(ii) We may write
0 =
k−1∑
i=0
s3i =
k−1∑
i=0
(−1)is3i+1 =
k−1∑
i=0
(−1)i(si + si−1)
3 =
= 3
k−1∑
i=0
(−1)isi−1s2i + 3
k−1∑
i=0
(−1)is2i−1si. (1)
On the other hand,
0 =
k−1∑
i=0
s3i =
k−1∑
i=0
(−1)i−1s3i−2 =
k−1∑
i=0
(−1)i(si − si−1)3 =
= 3
k−1∑
i=0
(−1)isi−1s2i − 3
k−1∑
i=0
(−1)is2i−1si. (2)
Adding (1) and (2) we obtain
6
k−1∑
i=0
si−1s
2
i = 0,
and subtracting (2) from (1) we have
6
k−1∑
i=0
s2i−1si = 0.
Since p > 3, (ii) follows.
Lemma 2.8 is proved.
Now we are ready to prove the main result.
Proof of main theorem. By using Lemma 2.4, it is sufficient to show that ak = ak+1 = bk =
= bk+1 = 0. We have
bk =
k−1∑
i=0
sk−1−isi−1si+1 =
k−1∑
i=0
s−(i+1)si−1si+1 =
k−1∑
i=0
(−1)isi−1s2i+1 =
=
k−1∑
i=0
(−1)isi−1(si−1 + si)
2 =
=
k−1∑
i=0
(−1)is3i−1 +
k−1∑
i=0
(−1)isi−1s2i + 2
k−1∑
i=0
(−1)is2i−1si,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
FIBONACCI LENGTHS OF ALL FINITE p-GROUPS OF EXPONENT p2 609
and the last three expressions vanish by Lemma 2.8. So bk = 0. Similarly,
bk+1 =
k∑
i=0
sk−isi−1si+1 =
k∑
i=0
s−isi−1si+1 =
k∑
i=0
(−1)i+1si−1sisi+1 =
=
k−1∑
i=0
(−1)i+1si−1sisi+1 =
k−1∑
i=0
(−1)i+1si−1si(si + si−1) =
= −
(
k−1∑
i=0
(−1)isi−1s2i +
k−1∑
i=0
(−1)is2i−1si
)
,
and the last two sums vanish by Lemma 2.8. On the other hand,
ak =
k−1∑
i=0
sk−1−i
(
si−1bi+1 +
(
si−1 + 1
2
)
si+1
)
=
=
k−1∑
i=0
sk−(i+1)
si−1 i∑
j=0
si−jsj−1sj+1 +
(
si−1 + 1
2
)
si+1
=
=
k−1∑
i=0
i∑
j=0
s−(i+1)si−1si−jsj−1sj+1 +
k−1∑
i=0
(
si−1 + 1
2
)
s−(i+1)si+1 =
=
k−1∑
i=0
i−1∑
j=0
s−i−1si−1si−jsj−1sj+1 +
1
2
k−1∑
i=0
(si−1 + 1)si−1s−(i+1)si+1,
and the first sum vanishes by Lemma 2.7(ii). For the second sum in the above expression, we have
k−1∑
i=0
(si−1 + 1)si−1s−(i+1)si+1 =
k−1∑
i=0
si−1si−1s−(i+1)si+1 +
k−1∑
i=0
si−1s−(i+1)si+1 =
=
k−1∑
i=0
si−2si−2s−isi +
k−1∑
i=0
(−1)isi−1s2i+1,
and the first sum vanishes by Lemma 2.7(i) and the second one is equal to bk which is zero. A similar
method may be used to prove ak+1 = 0. This completes the proof showing that k(G) = k(p2) for
all of groups G = Lα, where α = 0, 1, or non-residue modulo p.
Main theorem is proved.
Conjecture. For every p-group G of order pi, i = 5, 6, 7, k(G) = k(p2), where G is of
nilpotency class 3 and of exponent p2, for every odd prime p.
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Received 09.12.11,
after revision — 29.12.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 5
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| id | umjimathkievua-article-2444 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:34Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/de/bc5d4d6f71eae7d203f137b2ba5b52de.pdf |
| spelling | umjimathkievua-article-24442020-03-18T19:15:53Z Fibonacci lengths of all finite $p$-groups of exponent $p^2$ Довжини Фібоначчі для всіх скiнченних $p$-груп експоненти $p^2$ Ahmadi, B. Doostie, H. Ахмаді, Б. Доостіе, Г. The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent $p$ and the lengths depend on the Wall number $k(p)$. The p-groups of nilpotency class 3 and exponent $p$ were studied in 2004 also by Dikici. In the paper, we study all $p$-groups of nilpotency class 3 and exponent $p^2$. Thus, we complete the study of Fibonacci lengths of all $p$-groups of order $p^4$ by proving that the Fibonacci length is $k(p^2)$. Довжини Фібоначчі скінченних $p$-rpyn вивчалися Дікічі та співавторами з 1992 року. Всі групи, що розглядалися, були групами експоненти $p$, а всі довжини залежали від числа Уолла $k(p)$. $p$-Групи класу нільпотентності 3 i експоненти $p$ були також досліджені Дікічі у 2004 році. У даній статті ми вивчаємо всі $p$-групи класу нільпотентності 3 і експоненти $p^2$. Цим завершується дослідження довжини Фібоначчі всіх $p$-груп порядку $p^4$; при цьому доведено, що довжина Фібоначчі дорівнює $k(p^2)$. Institute of Mathematics, NAS of Ukraine 2013-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2444 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 5 (2013); 603–610 Український математичний журнал; Том 65 № 5 (2013); 603–610 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2444/1654 https://umj.imath.kiev.ua/index.php/umj/article/view/2444/1655 Copyright (c) 2013 Ahmadi B.; Doostie H. |
| spellingShingle | Ahmadi, B. Doostie, H. Ахмаді, Б. Доостіе, Г. Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title | Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title_alt | Довжини Фібоначчі для всіх скiнченних $p$-груп експоненти $p^2$ |
| title_full | Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title_fullStr | Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title_full_unstemmed | Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title_short | Fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| title_sort | fibonacci lengths of all finite $p$-groups of exponent $p^2$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2444 |
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