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Fermatean Fuzzy Maximal Subgroup and its Characterizations
A.Sheela Roselin
1
, D.Jayalakshmi
2
, G.Subbiah
3*
1
Research scholar, Reg.No.22123272092005, Department of Mathematics, Vivekananda
college, Agasteeswaram, Kanyakumari-629 701, Tamilnadu, India.
2
Associate Professor, Department of Mathematics, Vivekananda College, Agasteeswaram,
Kanyakumari-629 701, Tamilnadu, India.
3
Associate Professor, Department of Mathematics, Sri K.G.S Arts College, Srivaikuntam-628
619, Tamilnadu, India.
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.15020000037
Received: 18 February 2025; Accepted: 25 February 2026; Published: 05 March 2026
ABSTRACT
In this article, we investigate the concept of fermatean fuzzy characteristic subgroup of a group and fermatean
fuzzy normal subgroup. The level subset of fermatean fuzzy subgroup and its properties also defined. Finally,
the homomorphic image of fermatean fuzzy subgroup is also discussed.
Keywords: fuzzy set, fermatean fuzzy set, subgroup, characteristic, level subset, homomorphism.
AMS subject classification (2020): 03F55, 06D72, 08A72.
INTRODUCTION
The concept of fuzzy set is first introduced by L.A.Zadeh in 1965 [23]. After the tremendous development
intuitionistic fuzzy set [4], pythagoren fuzzy set [22], (3,2) fuzzy set[15], picture fuzzy set [10], Neutrosophic
fuzzy set [19] was studied. Recently fermatean fuzzy set [18] was studied by Yager and its application is briefly
explained by [18]. Due to the inadequacy of FST in the sense that it admits only the membership grade,
intuitionistic fuzzy sets (IFSs) was introduced [5, 6] by incorporating membership and non-membership grades
with the chance for hesitation margin. Various properties of IFSs were presented [ 4,5] and IFSs have been
applied in real-life problems [4, 5, 6]. Consequently, Biswas [8, 9] introduced intuitionistic fuzzy subgroups
(IFSGs) based on IFSs, Ahn et.al. [2] deliberated on sublattice of lattice of IFSGs of a group, some properties of
IFSGs were discoursed in [1]. In addition, Yuan et.al. [21] shared some light on the description of IFSGs, Bal
et.al. [7] presented a brief note of kernel subgroups on IFGs and derived some properties of IFGs. Senapati.T
[18] introduced fermatean fuzzy sets in 2020, an extension of intuitionistic fuzzy set (IFSs) and pythagorean
fuzzy set (PFSs). AVs and NAVs of fermatean fuzzy sets reveal their dependence on greater powers with sum
of cubes is less than 1. Ibrahim.H.Z introduced and applied n, m-rung orthopair fuzzy sets in MCDM.. The
concepts of fuzzy modules and fuzzy sub-modules was introduced by Nagotia and Ralescu [16 ] in 1975. The
concept of essential fuzzy modules was introduced by Hadi [13] in 2000. Using this idea. Abbas established the
concept of essential fuzzy sub-modules and uniform fuzzy modules in 2012. In this article, we investigate the
concept of fermatean fuzzy characteristic subgroup of a group and fermatean fuzzy normal subgroup. The level
subset of fermatean fuzzy subgroup and its properties also defined. Finally, the homomorphic image of fermatean
fuzzy subgroup is also discussed.
Preliminaries:
Throughout the article, the symbols and represent a non-empty set and a group respectively.
Definition-2.1(L.A.Zadeh): A fuzzy sub collection of is of the form
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where
is the membership grade of .
Definition-2.2(A.Rosenfeld): A fuzzy sub collection of is a fuzzy subgroup of if
(i)
(ii)
, .
where ‘’ is a minimum operation.
Furthermore,
, , where is the unit element of . Then
is the upper bound or tip of .
Definition-2.3(K.T. Atanassov):
An intuitionistic fuzzy set of is of the form
where
and
are the membership and non-membership grades of and
.
Definition-2.4(Yager’s): A fermatean fuzzy set of is of the form
where
and
are the membership and non-membership grades of and
.
Definition-2.5: Let
and
be fermatean fuzzy sets of . Then
(i)
and
, .
(ii)
and
, .
(iii)
(iv)
.
Definition-2.6: Let
be a fermatean fuzzy set on a group . Then is a fermatean fuzzy subgroup
of if
(i)
and
for all .
(ii)
and
for all .
Definition-2.7: If
and
are fermatean fuzzy subgroups of a group , then
is a fermatean fuzzy subgroup
of
if
, and
is a proper fermatean fuzzy subgroup of
if
, which gives
is strictly
contained in
.
Definition-2.8: Let
be a fermatean fuzzy subgroup of . Then, the level set of
is denoted by
and defined
by
. Also
is a subgroup of .
Definition-2.9: Suppose
be a fermatean fuzzy subgroup of . Then the strong
-cuts of
is denoted by
and defined by
. Also it is a subgroup of , where
.
Similarly, the weak
-cuts of
is denoted by
and defined by
. Also it is a subgroup of .
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Definition-2.10: A fermatean fuzzy subgroup
of commutative if
and
for all .
However, a fermatean fuzzy subgroup
is commutative if is a commutative group.
Definition-2.11:
Let
be a fermatean fuzzy subgroup of a fermatean fuzzy subgroup
of . Then
is a normal in
, denoted
as
if
and
and
for all .
Definition-2.12:
Assume and groups and is a homomorphism. Suppose
and
are fermatean fuzzy subgroups
of and respectively. Then induces a homomorphism from
to
that satisfies
(i)
and
for all
.
(ii)
and
for all
.
Main Results:
In the classical setting, we call a subgroup of a group is characteristic if
for every automorphism,
of , where
. Now, we establish the analogue of this idea as follows.
Definition-3.1: Let be fermatean fuzzy subgroup of , then is characteristic (-invariant) if
,
, for all for every automorphism of .
Hence for every of Aut.
Definition-3.2: Let be fermatean fuzzy subgroup in and is a mapping of . Define a fermatean fuzzy set
in by
,
, where
, for all in .
Proposition-3.3: Let be fermatean fuzzy subgroup in and . Suppose is an automorphism of for
for all , then
.
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Proof: For be fermatean fuzzy subgroup in and , assume that is characterized by
for all .
Then
,
.
Proposition-3.4: Let be fermatean fuzzy subgroup in and be an automorphism of . Then
is
fermatean fuzzy subgroup in .
Proof:For , we have
=
and
=
because is homomorphism.
Since is a fermatean fuzzy subgroup of , we have
and
.
Thus,
and
.
In addition,
=
and
=
.
Hence
is fermatean fuzzy subgroup in .
Theorem-3.5: Suppose is an automorphism and is a fuzzy subgroup in . Then
is a fermatean fuzzy subgroup in if and only if is a fermatean fuzzy subgroup in .
Proof:Suppose is a fermatean fuzzy subgroup in . Then using the procedure in the proposition-3.4, it is
certain that
is a fermatean fuzzy subgroup in .
Conversely, if
is a fermatean fuzzy subgroup in . Then
and
and
and
, .
Thus,
=
, .
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In addition,
=
and
=
, .
Hence is a fermatean fuzzy subgroup of .
Theorem-3.6: Every fermatean fuzzy characteristic subgroup if a fermatean fuzzy group is a fermatean fuzzy
normal subgroup.
Proof: Let and let be fermatean fuzzy characteristic subgroup.
We need to show that
and
.
Let us assume that is the automorphism of defined by
.
Now, since is fermatean fuzzy characteristic subgroup we have
.Then
and
Hence
is a normal subgroup.
Remark-3.7: Let be fermatean fuzzy subgroups in in which
(i) If is a fermatean fuzzy characteristic subgroup of and is fermatean fuzzy characteristic
subgroup of , then is a fermatean fuzzy characteristic subgroup of .
(ii) If is a fermatean fuzzy normal subgroup of and is fermatean fuzzy normal subgroup of , then
is a fermatean fuzzy normal subgroup of .
Proposition-3.8: If is fermatean fuzzy subgroup in and is fermatean fuzzy characteristic subgroup of ,
then is a characteristic subgroup of . In addition, is a
characteristic subgroup of G.
Proof:It is certain that
is a subgroup of .
To explain that
is characteristic in , it is sufficient to show that
, Aut
.
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Let Aut
, then
and
since is fermatean fuzzy characteristic
subgroup of .
Let
, then
and
, which gives
and
.
So
. Hence
which concludes the result.
In addition, because is a fermatean fuzzy characteristic subgroup of and
is a
characteristic subgroup of , it shows
is a characteristic subgroup of
.
Theorem-3.9: Suppose is finite and is a fermatean fuzzy characteristic subgroup of in . Then
and
are characteristic subgroups of .
Proof:It is obvious that,
is a subgroup of . Further we will clear that
is characteristic in by
showing that
, Aut
.
For Aut
, we get
and
, because is a fermatean fuzzy characteristic
subgroup.
Let
. Then
and
, so
. Thus
. Hence,
is a
characteristic subgroup of .
Similarly,
is a characteristic subgroup of .
Remark-3.10: Because is a fermatean fuzzy characteristic subgroup of , and
and
are
characteristic subgroups of , it follows that
is a characteristic subgroup of
and
is a
characteristic subgroup of
.
Theorem-3.11: Let be a fermatean fuzzy subgroup of in , where is finite. If
and
are
characteristic subgroups of , then is fermatean fuzzy characteristic subgroup of .
Proof: Since is finite,
. Let
with
and
.
By assumption,
is a characteristic subgroup of ,
Let Aut
. Because
and
, then
.
Moreover, , we have
.
Since
.
Thus
and hence is fermatean fuzzy characteristic subgroup of .
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In what follows we validate that theorem-3.9 and theorem-3.11 are still true by discarding the finite order of .
Theorem-3.12:Suppose is fermatean fuzzy subgroup of in . Then the following statements are the same;
(i) is fermatean fuzzy characteristic subgroup of .
(ii)
and
are fermatean fuzzy characteristic subgroups in .
Proof:In this proof (i)(ii)
Let
, Aut
and
.
Assume that is fermatean fuzzy characteristic subgroup of . Then
and
.
Thus
and so,
.
But
by symmetry.
Let
and where in
.
Then,
and
.
, therefore
.
Hence
.
Thus,
is a fermatean fuzzy characteristic subgroup in .
Similarly,
is a fermatean fuzzy characteristic subgroup in .
Next, we prove (ii)(i).
Let , Aut
and
,
.
Then
and
for
.
By hypothesis
, so
and hence
and
.
Let
and
.
Assume
, then
.
Because is injective, we get
, which is a contradiction.
Thus,
and
.
is fermatean fuzzy characteristic subgroup of .
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Theorem-3.13: Let be a fermatean fuzzy subgroup that is normal in a fermatean fuzzy subgroup in and
assume is an automorphism of which leaves an invariant subgroup
. Then induces an automorphism
of defined by
, .
Proof:To start with, we confirm that
is well defined.
Suppose such that . Then, it is sufficient to prove that
.
Since , we get
and
and
.
Similarly,
and
.
and
.
.
,
.
By assumption,
, then
,
Thus,
and
.
Now, let , then
min
min
and
max
max
.
Thus,
and
.
Similarly,
and
.
Since
is arbitrary, then
and
.
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.
Thus, is well defined.
In addition, let . Because is homomorphism,
and so,
.
Hence
and
.
Therefore,
is homomorphism.
Corollary-3.14: Using the theorem-3.13 is an automorphism if is finite and is an automorphism.
Proof:Let
, for .
Then
, which gives
.
Hence , since
is injective.
Thus,
.
and
.
.
So,
and
, since
.
Thus,
and
.
Hence is injective, which completes the proof.
Corollary-3.15:Let a fermatean fuzzy subgroup be normal in a fermatean fuzzy subgroup in . If is an
automorphism of and
, then induces an automorphism of
of defined by
, for
all .
Proof:Let . Then
.
,
,
.
,
,
.
,
,
.
,
,
.
,
.
Thus ,
is well defined and injective. Certainly,
maps onto itself. Because
is homomorphism, then
the proof is completed by the similar wave length of the proof
theorem-3.13.
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Theorem-3.16: Suppose is a homomorphism, and are fermatean fuzzy subgroups of and
respectively. Then
implies
and
, for all .
Proof:Let
and .
Then,
min
and
max
.
By inversely, let
and
,for all .
Then,
max
max
and
min
min
for all .
Hence,
.
CONCLUSION
In this article, the concepts of fermatean fuzzy characteristic maximal subgroups were shown. Various results
concerning fermatean fuzzy characteristic subgroups were obtained and explained in detail.
ACKNOWLEDGEMENT
The authors are highly greatful to the refees for their valuable comments and suggestions to improve the paper.
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