Lehmer sequences in finite groups
We study the Lehmer sequences modulo $m$. Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a 2-generator group $G$ for a generating pair $(x, y) \in G$ and examine the lengths of the periods of these orbits. Furthermore, we obtain the Lehmer lengths and the basic Lehmer lengths of...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507700574552064 |
|---|---|
| author | Deveci, Ö. Karaduman, E. Девесі, О. Карадуман, Е. |
| author_facet | Deveci, Ö. Karaduman, E. Девесі, О. Карадуман, Е. |
| author_sort | Deveci, Ö. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:29:16Z |
| description | We study the Lehmer sequences modulo $m$. Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a
2-generator group $G$ for a generating pair $(x, y) \in G$ and examine the lengths of the periods of these orbits. Furthermore, we obtain the Lehmer lengths and the basic Lehmer lengths of the Fox groups $G_{1,t}$ for $t \geq 3$. |
| first_indexed | 2026-03-24T02:13:29Z |
| format | Article |
| fulltext |
UDC 512.5
Ö. Deveci (Kafkas Univ., Turkey),
E. Karaduman (Atatürk Univ., Turkey)
LEHMER SEQUENCES IN FINITE GROUPS*
ПОСЛIДОВНОСТI ЛЕМЕРА У СКIНЧЕННИХ ГРУПАХ
We study the Lehmer sequences modulo m. Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a
2-generator group G for a generating pair (x, y) \in G and examine the lengths of the periods of these orbits. Furthermore,
we obtain the Lehmer lengths and the basic Lehmer lengths of the Fox groups G1,t for t \geq 3.
Вивчаються послiдовностi Лемера за модулем m. Крiм того, визначено поняття орбiти Лемера та базової орбiти
Лемера двогенераторної групи G для породжуючої пари (x, y) \in G та дослiджено довжини перiодiв для цих орбiт.
Також встановлено довжини Лемера та базовi довжини Лемера для груп Фокса G1,t при t \geq 3.
1. Introduction and preliminaries. The Lehmer sequence U = U (L,M) = \{ Un\} \infty 0 is the sequence
of integers which is defined by integer constants L, M, U0 = 0, U1 = 1 and the recurrence
Un =
\left\{ LUn - 1 - MUn - 2 for n \mathrm{o}\mathrm{d}\mathrm{d},
Un - 1 - MUn - 2 for n \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},
(1)
where LM \not = 0 and K = L - 4M \not = 0. The sequence U is called a Lehmer sequence and Un is
a Lehmer number. For more information on this sequence, see [6]. The Lehmer numbers and their
properties have been studied by some authors (see, for example, [5, 7, 8]).
It is well-known that a sequence 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 the period of the
sequence. A sequence is simply periodic with period k if the first k elements in the sequence form a
repeating subsequence.
The study of Fibonacci sequences in groups began with the earlier work of Wall [9]. In the mid
eighties, Wilcox extended the problem to Abelian groups [10]. Campbell, Doostie and Robertson [1]
expanded the theory to some simple groups. 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. Deveci
and Karaduman [4] defined the generalized order-k Pell sequences in finite groups and obtained the
periods of the generalized order-k Pell sequences in dihedral groups Dn. Deveci [3] expanded the
concept to the Pell – Padovan sequence and the Jacobsthal – Padovan sequence. Now we extend the
concept to the Lehmer sequences.
In this paper, the usual notation p is used for a prime number and the notation
\bigl\{
UM,L
\bigr\}
is used
for the Lehmer sequence U.
2. The Lehmer sequences modulo \bfitalpha . Reducing the Lehmer sequence by a modulus \alpha , we can
get a repeating sequence, denoted by\bigl\{
UM,L(\alpha )
\bigr\}
=
\Bigl\{
UM,L
0 (\alpha ), UM,L
1 (\alpha ), UM,L
2 (\alpha ), . . . , UM,L
i (\alpha ), . . .
\Bigr\}
,
where UM,L
i (\alpha ) = UM,L
i (\mathrm{m}\mathrm{o}\mathrm{d} \alpha ). It has the same recurrence relation as in (1).
* This paper was supported by the Commission for the Scientific Research Projects of Kafkas University (Project
№. 2011-FEF-26).
c\bigcirc Ö. DEVECI, E. KARADUMAN, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2 175
176 Ö. DEVECI, E. KARADUMAN
Theorem 2.1. The sequence
\bigl\{
UM,L(\alpha )
\bigr\}
is simply periodic if M = \pm 1, and is periodic otherwise.
Proof. The sequence repeats since there are only a finite number \alpha 2 of pairs of terms possible,
and the recurrence of a pair results in recurrence of all following terms, which impliest that the
sequence
\bigl\{
UM,L(\alpha )
\bigr\}
is periodic. From definition of the Lehmer sequence we have
MUn - 2 =
\left\{ LUn - 1 - Un for n \mathrm{o}\mathrm{d}\mathrm{d},
Un - 1 - Un for n \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},
so if UM,L
i+1 (\alpha ) \equiv UM,L
j+1 (\alpha ), UM,L
i (\alpha ) \equiv UM,L
j (\alpha ) and M = \pm 1, then UM,L
i - j+1(\alpha ) \equiv UM,L
1 (\alpha ) and
UM,L
i - j (\alpha ) \equiv UM,L
0 (\alpha ), which implies that the sequence
\bigl\{
UM,L(\alpha )
\bigr\}
is simply periodic.
Let kM,L(\alpha ) denote the smallest period of the sequence
\bigl\{
UM,L(\alpha )
\bigr\}
, called the period of the
Lehmer sequences modulo \alpha .
Example. We have
\bigl\{
U1,5 (7)
\bigr\}
= \{ 0, 1, 1, 4, 3, 4, 1, 1, 0, 6, 6, 3, 4, 3, 6, 6, 0, 1, 1, 4, . . .\} . So, we
get k1,5 (7) = 16.
Theorem 2.2. If m =
\prod t
i=1
peii , t \geq 1, where pi are distinct primes, then kM,L(m) =
= lcm
\bigl[
kM,L(peii )
\bigr] \Bigl(
where the least common multiple of kM,L(pe11 ), kM,L(pe22 ), . . . , kM,L(pett )
is denoted by lcm
\bigl[
kM,L(peii )
\bigr] \Bigr)
.
Proof. The statement, “kM,L (peii ) is the length of the period of
\bigl\{
UM,L (peii )
\bigr\}
”, implies that the
sequence
\bigl\{
UM,L (peii )
\bigr\}
repeats only after blocks of length u \cdot kM,L (peii ) , u \in N, and the statement,
“kM,L (m) is the length of the period
\bigl\{
UM,L (m)
\bigr\}
”, implies that
\bigl\{
UM,L (peii )
\bigr\}
repeats after
kM,L(m) terms for all values i. Thus, kM,L(m) is of the form u \cdot kM,L(peii ) for all values of i, and
since any such number gives a period of
\bigl\{
UM,L (m)
\bigr\}
. Then we get that kM,L(m) = lcm
\bigl[
kM,L(peii )
\bigr]
.
Theorem 2.3. If kM,L(p2) \not = kM,L(p) and M = \pm 1, then kM,L(p2) = p.kM,L(p).
Proof. Let kM,L(p2) \not = kM,L(p) and M = \pm 1, then the sequence \{ UM,L\} is
UM,L
0 = 0, UM,L
1 = 1, . . . ,
UM,L
kM,L(p)
= \lambda 1.p, UM,L
kM,L(p)+1
= \lambda 2.p+ 1, . . . ,
UM,L
2.kM,L(p)
= \lambda 1.2p, UM,L
2.kM,L(p)+1
= \lambda 2.2p+ 1, . . . ,
UM,L
p.kM,L(p)
= \lambda 1.p
2, UM,L
p.kM,L(p)+1
= \lambda 2.p
2 + 1, . . . ,
where \lambda 1, \lambda 2 \in N such that p \nmid gcd(\lambda 1, \lambda 2)
\bigl(
where by p \nmid \itg \itc \itd (\lambda 1, \lambda 2) we mean that p not di-
vides greatest common divisor \lambda 1 and \lambda 2
\bigr)
. Since the elements succeeding UM,L
p.kM,L(p)
\equiv 0 and
UM,L
p.kM,L(p)+1
\equiv 1, the cycles begins again with the p2
nd
element, i.e., UM,L
p.kM,L(p)
\equiv UM,L
0 and
UM,L
p.kM,L(p)+1
\equiv UM,L
1 . Then we get that kM,L(p2) = p.kM,L(p).
Conjecture 2.1. (i) If p \not = 2, kM,L(pt+1) \not = kM,L(pt), t \geq 1, and M = \pm 1, then kM,L(pt+1) =
= p.kM,L(pt).
(ii) If kM,L(2t+1) \not = kM,L(2t), t \geq 2, and M = \pm 1, then kM,L(2t+1) = 2.kM,L(2t).
3. The Lehmer length and the basic Lehmer length of generating pairs in groups. Let G be
a group and let x, y \in G. If every element of G can be written as a word
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
LEHMER SEQUENCES IN FINITE GROUPS 177
xu1 yu2 xu3 yu4 . . . xum - 1 yum , (2)
where ui \in Z, 1 \leq i \leq m, then we say that x and y generate G and that G is a 2-generator group.
Let G be a finite 2-generator group and X be the subset of G\times G such that (x, y) \in X if, and only
if, G is generated by x and y. We call (x, y) a generating pair for G.
Definition 3.1. For a generating pair (x, y) \in G, we define the Lehmer orbit UM,L
x,y (G) = \{ xi\}
as follows:
x0 = x, x1 = y, xi+1 =
\left\{ (xi - 1)
- M (xi)
L for i even,
(xi - 1)
- M (xi) for i odd,
i \geq 1.
Theorem 3.1. A Lehmer orbit UM,L
x,y (G) of a finite group is simply periodic if M = \pm 1, and is
periodic otherwise.
Proof. Let n be the order of G. Since there are n2 distinct 2-tuples of elements of G, at least one
of the 2-tuples appears twice in a Lehmer orbit of G. Thus, the subsequence following this 2-tuples.
Because of the repeating, the Lehmer orbit is periodic.
Since the Lehmer orbit is periodic, there exist natural numbers u and v, with u > v, such that
xu+1 = xv+1, xu+2 = xv+2.
By the defining relation of the Lehmer orbit, we know that
(xu)
- M =
\left\{ (xu+2)(xu+1)
- L for v odd,
(xu+2)(xu+1)
- 1 for v even,
and (xv)
- M =
\left\{ (xv+2)(xv+1)
- L for v odd,
(xv+2)(xv+1)
- 1 for v even.
Hence, xu = xv for M = \pm 1, and it then follows that
xu - v = xv - v = x0, xu - v+1 = xv - v+1 = x1.
Thus, the Lehmer orbit UM,L
x,y (G) is simply periodic for M = \pm 1.
In this paper, we denote the length of the period of the Lehmer orbit UM,L
x,y (G) by \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G)
and we call the Lehmer length of G with respect to generating pair (x, y) and integer constants L,M.
Lemma 3.1. If M = \pm 1 and the Lehmer orbit UM,L
x,y (G) of (x, y) \in X has length n1, then for
any i, 0 \leq i \leq n1 - 1, we have (xi, xi+1) \in X. Also we have UM,L
x,y (G) = UM,L
xi,yi(G).
Proof. We will use the induction method on i to show (xi, xi+1) \in X. The case i = 0 is trivially
true. Suppose by way of inductive hypothesis that (xk, xk+1) \in X and consider (xk+1, xk+2). Now
(xk)
- M =
\left\{ (xk+2)(xk+1)
- L for k odd,
(xk+2)(xk+1)
- 1 for k even,
so, since every element of G has an expression of the form (2) with xk = x, xk+1 = y, we see that,
on replacing (xk)
- M by \left\{ (xk+2)(xk+1)
- L for k odd,
(xk+2)(xk+1)
- 1 for k even,
every element of G is generated by xk+1 and xk+2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
178 Ö. DEVECI, E. KARADUMAN
Finally suppose UM,L
x,y (G) = \{ xi\} and UM,L
r,s (G) = \{ bi\} . Then again an inductive argument
proves that if x0 = bj , x1 = bj+1, then UM,L
x,y (G) = UM,L
r,s (G).
For suppose xi = bi+j , i < t. Then xt = (xt - 2)
- 1(xt - 1)
2 = (bt - 2+j)
- 1(bt - 1+j)
2 = bj+t and
the result is proved.
Lemma 3.1 gives immediately the following theorem.
Theorem 3.2. If M = \pm 1 and G is a finite group, then X partitioned by and the Lehmer orbits
UM,L
x,y (G) for (x, y) \in X.
To examine the concept more fully we study the action of the automorphism group \mathrm{A}\mathrm{u}\mathrm{t}G of G
on X and on the Lehmer orbits UM,L
x,y (G), (x, y) \in X. Now \mathrm{A}\mathrm{u}\mathrm{t}G consist of all isomorphisms \theta :
G \rightarrow G and if \theta \in \mathrm{A}\mathrm{u}\mathrm{t}G and (x, y) \in X, then (x\theta , y\theta ) \in X.
For a subset A \subseteq G and \theta \in \mathrm{A}\mathrm{u}\mathrm{t}G the image of A under \theta is A\theta = \{ a\theta : a \in A\} .
Lemma 3.2. Let (x, y) \in X and \theta \in \mathrm{A}\mathrm{u}\mathrm{t} G. If M = \pm 1, then UM,L
x,y (G)\theta = UM,L
x\theta ,y\theta (G).
Proof. Let UM,L
x,y (G) = \{ xi\} . Now \{ xi\} \theta = \{ xi\theta \} and since\bigl(
(xi - 1)
- M (xi)
L
\bigr)
\theta = (xi - 1)
- M\theta (xi)
L\theta and
\bigl(
(xi - 1)
- M (xi)
\bigr)
\theta = (xi - 1)
- M\theta (xi)\theta
the result follows.
If M = \pm 1 and n of the elements of \mathrm{A}\mathrm{u}\mathrm{t}G map UM,L
x,y (G) into itself. Then there are | \mathrm{A}\mathrm{u}\mathrm{t}G| /n
distinct Lehmer orbits UM,L
x\theta ,y\theta (G) for \theta \in \mathrm{A}\mathrm{u}\mathrm{t}G.
Definition 3.2. For a generating pair (x, y) \in X and M = \pm 1, we define the basic Lehmer
orbits UM,L
x,y (G) of basic length m to be the sequence \{ xi\} of elements of G such that
x0 = x, x1 = y, xi+1 =
\left\{ (xi - 1)
- M (xi)
L for i even,
(xi - 1)
- M (xi) for i odd,
i \geq 1,
where m \geq 1 is least integer with
x0 = xm\theta , x1 = xm+1\theta ,
for some \theta \in \mathrm{A}\mathrm{u}\mathrm{t} G.
Since xm, xm+1 generate G, it follows that \theta is uniquely determined.
In this paper, we denote the length of the period of the the basic Lehmer orbit UM,L
x,y (G) by
\mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G) and we call the basic Lehmer length of G with respect to generating pair (x, y) and
integer constants L,M.
From the definitions it is clear that the Lehmer lengths and the basic Lehmer lengths of a group
depend on the chosen generating set and the order in which the assignments of x0, x1 are made.
Theorem 3.3. Let G be a finite group and (x, y) \in X. If M = \pm 1, the orbit UM,L
x,y (G) has
length n1 and the basic orbit UM,L
x,y (G) has length m1, then m1 divides n1 and there n1/m1 elements
of \mathrm{A}\mathrm{u}\mathrm{t}G which map UM,L
x,y (G) into itself.
Proof. Since UM,L
x,y (G) = UM,L
x,y (G) \cup UM,L
x\theta ,y\theta (G) \cup UM,L
x\theta 2,y\theta 2
(G) \cup . . . and \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G) =
= \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x\theta ,y\theta (G) we have n1 = m1 \cdot \lambda , where \lambda is order of automorphism \theta \in \mathrm{A}\mathrm{u}\mathrm{t}G. Clearly
1, \theta , \theta 2, . . . , \theta \lambda - 1 map UM,L
x,y (G) into itself.
4. The Lehmer lengths and the basic Lehmer lengths of the Fox groups. The Fox groups
G1,t, are finite metacyclic groups of order | t - 1| 3, having generators of order (t - 1)2 (see [2]). They
are presented by
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
LEHMER SEQUENCES IN FINITE GROUPS 179\bigl\langle
x, y : xy = ytx, yx = xty
\bigr\rangle
.
The relations of G1,t imply the relation xt - 1 = y1 - t.
In this section, we obtain the Lehmer lengths and the basic Lehmer lengths of G1,t for M = \pm 1
and t \geq 3.
Theorem 4.1. (i) Let t = 3, then tree cases occur:
(1) If M = 1 and L is an integer such that L \not = 0, then
\mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) =
\left\{
4, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
3, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
8, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
6, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4)
and
\mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) =
\left\{
2, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
1, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
2, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
2, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4).
(2) If M = - 1 and L is an integer such that L > 0, then
\mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) =
\left\{
8, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
3, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
4, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
3, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4)
and
\mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) =
\left\{
2, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
1, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
2, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
1, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4).
(3) If M = - 1 and L is an integer such that L < 0, then
\mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) =
\left\{
8, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
3, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
4, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
6, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
180 Ö. DEVECI, E. KARADUMAN
and
\mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) =
\left\{
2, L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
1, L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
2, L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
2, L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4).
(ii) Let t \geq 4, then two cases occur:
(1\prime ) If kM,L
\bigl(
(t - 1)2
\bigr)
= kM,L(t - 1), then \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = kM,L(t - 1).
(2\prime ) If kM,L
\bigl(
(t - 1)2
\bigr)
\not = kM,L(t - 1), then \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = kM,L
\bigl(
(t - 1)2
\bigr)
and
\mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = kM,L(t - 1).
Proof. i (1) Let M = 1 and L is an integer such that L \not = 0.
If L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, yx, y - 1, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 4 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 2 since x\theta = yx and y\theta = y - 1, where \theta is a
outher automorphism of order 2.
If L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, yx, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 3 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 1 since x\theta = yx and y\theta = x, where \theta is a
outher automorphism of order 3.
If L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, yx, y, x - 1, y, xy, y, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 8 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 2 since x\theta = xy and y\theta = y, where \theta is a
outher automorphism of order 4.
If L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, yx, x - 1, y - 1, xy, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 6 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 2 since x\theta = y - 1 and y\theta = xy, where \theta is a
outher automorphism of order 3.
(2) Let M = - 1 and L is an integer such that L > 0.
If L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, y, x - 1, y, yx, y, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 8 and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 2 since x\theta = yx and y\theta = y, where \theta is a
outher automorphism of order 4.
If L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
LEHMER SEQUENCES IN FINITE GROUPS 181
x, y, xy, y - 1, x, y, . . . .
So we get LenU - 1,L
x,y (G1,3) = 4 and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 2 since x\theta = xy and y\theta = y - 1, where \theta is
a outher automorphism of order 2.
If L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4) or L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 3 and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 1 since x\theta = xy and y\theta = x, where \theta is a
outher automorphism of order 3.
(3) Let M = - 1 and L is an integer such that L < 0.
If L \equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, y, x - 1, y, yx, y, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 8 and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 2 since x\theta = yx and y\theta = y, where \theta is a
outher automorphism of order 4.
If L \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 3 and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,3) = 1 since x\theta = xy and y\theta = x, where \theta is a
outher automorphism of order 3.
If L \equiv 2 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, y - 1, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 4 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 2 since x\theta = xy and y\theta = y - 1, where \theta is a
outher automorphism of order 2.
If L \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4), then the Lehmer orbit is
x, y, xy, x - 1, y - 1, yx, x, y, . . . .
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 6 and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,3) = 2 since x\theta = y - 1 and y\theta = yx, where \theta is a
outher automorphism of order 3.
ii (1\prime ) The proof is similar to the proof of the Theorem 3.1 in [2] and is omitted.
(2\prime ) If kM,L
\bigl(
(t - 1)2
\bigr)
\not = kM,L(t - 1), then there are 3 subcases:
Case 1. If M = 1 and L is an integer such that L > 0 and M = - 1 and L is an integer such
that L < 0, then the Lehmer orbit UM,L
x,y (G1,t) is
x0 = x, x1 = y, . . . ,
xkM,L(t - 1) = x - t2+3t - 1, xkM,L(t - 1)+1 = y, . . . ,
x(t - a)kM,L(t - 1) = x( - a+1)t+a, x(t - a)kM,L(t - 1)+1 = y, . . . ,
x(t - 1)kM,L(t - 1) = xkM,L((t - 1)2) = x, x(t - 1)kM,L(t - 1)+1 = x
kM,L
\bigl(
(t - 1)2
\bigr)
+1
= y, . . . ,
where 2 \leq a \leq t - 2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
182 Ö. DEVECI, E. KARADUMAN
So we get \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = kM,L
\bigl(
(t - 1)2
\bigr)
and \mathrm{L}\mathrm{e}\mathrm{n}UM,L
x,y (G1,t) = kM,L(t - 1) since x\theta =
= x - t+2 and y\theta = y, where \theta is the inner automorphism induced by conjugation by yt - 2.
Case 2. If M = 1 and L is an integer such that L < 0, then the Lehmer orbit U1,L
x,y (G1,t) is
x0 = x, x1 = y, . . . ,
xk1,L(t - 1) = x(t)
t - 2
, xk1,L(t - 1)+1 = y, . . . ,
x(t - a)k1,L(t - 1) = x(t)
a - 1
, x(t - a)k1,L(t - 1)+1 = y, . . . ,
x(t - 1)k1,L(t - 1) = x
k1,L
\bigl(
(t - 1)2
\bigr) = x, x(t - 1)k1,L(t - 1)+1 = x
k1,L
\bigl(
(t - 1)2
\bigr)
+1
= y, . . . ,
where 2 \leq a \leq t - 2.
So we get \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,t) = k1,L
\Bigl(
(t - 1)2
\Bigr)
and \mathrm{L}\mathrm{e}\mathrm{n}U1,L
x,y (G1,t) = k1,L (t - 1) since x\theta =
= xt and y\theta = y, where \theta is the inner automorphism induced by conjugation by y.
Case 3. If M = - 1 and L is an integer such that L > 0, then the Lehmer orbit U1,L
x,y (G1,t) is
x0 = x, x1 = y, . . . ,
xk1,L(t - 1) = x(t)
t - 2
, xk1,L(t - 1)+1 = y - t2+3t - 1, . . . ,
x(t - a)k1,L(t - 1) = x(t)
a - 1
, x(t - a)k1,L(t - 1)+1 = y( - a+1)t+a, . . . ,
x(t - 1)k1,L(t - 1) = xk1,L((t - 1)2) = x, x(t - 1)k1,L(t - 1)+1 = xk1,L((t - 1)2)+1 = y, . . . ,
where 2 \leq a \leq t - 2.
So we get \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,t) = k - 1,L
\bigl(
(t - 1)2
\bigr)
and \mathrm{L}\mathrm{e}\mathrm{n}U - 1,L
x,y (G1,t) = k - 1,L(t - 1) since
x\theta = xt and y\theta = y - t+2, where \theta is a outher automorphism of order t - 1.
References
1. Campbell C. M., Doostie H., Robertson E. F. Fibonacci length of generating pairs in groups // Appl. Fibonacci
Numbers. – 1990. – 3. – P. 27 – 35.
2. Campbell C. M., Campbell P. P., Doostie H., Robertson E. F. Fibonacci lengths for certain metacyclic groups //
Algebra colloq. – 2004. – 11, № 2. – P. 215 – 229.
3. Deveci O. The Pell – Padovan sequences and the Jacobsthal – Padovan sequences in finite groups // Util. Math. –
2015. – 98. – P. 257 – 270.
4. Deveci O., Karaduman E. The Pell sequences in finite groups // Util. Math. – 2015. – 96. – P. 263 – 276.
5. Fuller A. T. The period of pseudo-random numbers generated Lehmer’s congruential method // Comput. J. – 1976. –
19, № 2. – P. 173 – 177.
6. Lehmer D. H. An extended theory of Lucas functions // Ann. Math. – 1930. – 31, № 2. – P. 419 – 448.
7. Phong B. M. On generalized Lehmer sequences // Acta Math. Hung. – 1991. – 57, № 3-4. – P. 201 – 211.
8. Rotkiewicz A. On strong Lehmer pseuduprimes in the case of negative discriminant in arithmetic progressions // Acta
arithm. – 1994. – 68, № 2. – P. 145 – 151.
9. Wall D. D. Fibonacci series modulo m // Amer. Math. Mon. – 1960. – 67. – P. 525 – 532.
10. Wilcox H. J. Fibonacci sequences of period n in groups // Fibonacci Quart. – 1986. – 24. – P. 356 – 361.
Received 17.05.13
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 2
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| id | umjimathkievua-article-1832 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:29Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d3/8aef717367e454af4e27a01e3c66d9d3.pdf |
| spelling | umjimathkievua-article-18322019-12-05T09:29:16Z Lehmer sequences in finite groups Послiдовностi Лемера у скiнченних групах Deveci, Ö. Karaduman, E. Девесі, О. Карадуман, Е. We study the Lehmer sequences modulo $m$. Moreover, we define the Lehmer orbit and the basic Lehmer orbit of a 2-generator group $G$ for a generating pair $(x, y) \in G$ and examine the lengths of the periods of these orbits. Furthermore, we obtain the Lehmer lengths and the basic Lehmer lengths of the Fox groups $G_{1,t}$ for $t \geq 3$. Вивчаються послiдовностi Лемера за модулем $m$. Крiм того, визначено поняття орбiти Лемера та базової орбiти Лемера двогенераторної групи $G$ для породжуючої пари $(x, y) \in G$ та дослiджено довжини перiодiв для цих орбiт. Також встановлено довжини Лемера та базовi довжини Лемера для груп Фокса $G_{1,t}$ при $t \geq 3$. Institute of Mathematics, NAS of Ukraine 2016-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1832 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 2 (2016); 175-182 Український математичний журнал; Том 68 № 2 (2016); 175-182 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1832/814 Copyright (c) 2016 Deveci Ö.; Karaduman E. |
| spellingShingle | Deveci, Ö. Karaduman, E. Девесі, О. Карадуман, Е. Lehmer sequences in finite groups |
| title | Lehmer sequences in finite groups |
| title_alt | Послiдовностi Лемера у скiнченних групах |
| title_full | Lehmer sequences in finite groups |
| title_fullStr | Lehmer sequences in finite groups |
| title_full_unstemmed | Lehmer sequences in finite groups |
| title_short | Lehmer sequences in finite groups |
| title_sort | lehmer sequences in finite groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1832 |
| work_keys_str_mv | AT devecio lehmersequencesinfinitegroups AT karadumane lehmersequencesinfinitegroups AT devesío lehmersequencesinfinitegroups AT karadumane lehmersequencesinfinitegroups AT devecio poslidovnostilemerauskinčennihgrupah AT karadumane poslidovnostilemerauskinčennihgrupah AT devesío poslidovnostilemerauskinčennihgrupah AT karadumane poslidovnostilemerauskinčennihgrupah |