ACADEMIC RESEARCH IN MODERN SCIENCE
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37
NORMAL SUBGROUPS OF INDEX 8 IN THE GROUP
REPRESENTATION OF THE CAYLEY TREE
G'aybullayev Dilshodbek Erkin òg'li
gaybullayevdilshodbek0099@gamil.com
Karshi state university,
Xalilov Akbar Zamirovich
Karshi state university
https://doi.org/10.5281/zenodo.13950089
Many papers and books have been written on the theory of groups.
However, there are still unsolved problems, many of which arise in fields such as
physics, biology, and other natural sciences. For example, when the
configuration of a physical system is defined on a lattice (which can be viewed as
a graph of a group), the configuration may be interpreted as a function defined
over the lattice. Numerous studies have explored different types of partitions in
groups (lattices) (see for example [1]-[3]).
Let
k
G
represent a free product of
1
k
cyclic groups of order two, with
generators
1
2
1
,
,
,
k
a a
a
. The group
G
is assumed to have a finite number of
generators of order two, and let
r
represent the minimum number of such
generators for
G
. Without loss of generality, we can designate these generators
as
1
2
,
,
,
r
b b
b
. Additionally, let
1
e
denote the identity element of
G
. We then
define a homomorphism from
k
G
onto
G
.
Let
1
2
{ ,
,
,
}
n
n
A A
A
be a partition of the set
0
\
k
N
A
, where
0
0 |
|
1
A
k
n
. The homomorphism
1
2
1
1
1
:{ ,
,
,
}
{ , ,
,
}
n
k
m
u
a a
a
e b
b
is then
given by the following expression:
1
0
____
,
,
( )
,
,
,
1,
i
n
j
i
j
e if x
a i
A
u x
b if x
a i
A j
n
(1)
For any element
b
G
, we define
1
2
[ ,
,
,
]
b
m
R b b
b
as a representation of the
word
b
in terms of the generators
1
2
,
,
,
r
b b
b
, with
r
m
. The homomorphism
:
n
G
G
is defined by the formula:
ACADEMIC RESEARCH IN MODERN SCIENCE
International scientific-online conference
38
1
1
1
,
( )
,
,
1,
[ ,...,
],
,
1,
i
n
i
i
b
r
i
e if x
e
x
b if x
b i
r
R b
b
if x
b i
r
(2)
Theorem.
For the group
k
G
following statement is hold
{
| |
|
:
| 10}
k
k
H
H is a normal subgroup of G with G
H
0 1 2
(5)
10
1
2
0
{
(
) |
,
}.
B B B
k
H
R
B B is a partition of the set N
B
Reference:
1. Ganikhodjaev, N.N., Rozikov, U.A., (1997), Description of periodic extreme
Gibbs measures of some lattice model on the Cayley tree, Theor.Math.Phys. 111,
pp. 480-486.
2. U.A., Rozikov, F.H., Haydarov., (2014), Normal subgroups of finite index for the
group represantation of the Cayley tree, TWMS Jour.Pure.Appl.Math. 5, pp. 234-
240.
3. U.A., Rozikov., (2013) Gibbs measures on a Cayley trees, World Sci. Pub,
Singapore. Normal Subgroups of Index 8 in the Group Representation of the
Cayley Tree
