AFRICAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Imhotep Mathematical Proceedings
Volume 4, Numéro 1, (2017),
pp. 27 - 33.
Classifying a class of the fuzzy subgroups of the alternating groups An
M.E. Ogiugo
Department of Mathematics,
University of Ibadan, Ibadan, Abstract
Nigeria.
The aim of this paper is to classify the fuzzy subgroups of the alternating
ekpenogiugo@gmail.com group. First, an equivalence relation on *the set of all fuzzy subgroups of
a group G is defined. Without any equivalence relation on fuzzy subgroups
M. Enioluwafe of group G, the number of fuzzy subgroups is infinite, even for the trivial
group. Explicit formulae for the number of distinct fuzzy subgroup of finite
Department of Mathematics, alternating group are obtained in the particular case n = 5. Some inequalities
University of Ibadan, Ibadan, satisfied by this number are also established for n ≥ 5
Nigeria.
michael.enioluwafe@gmail.com
Proceedings of the 6th annual workshop on
CRyptography, Algebra and Geometry http://imhotep-journal.org/index.php/imhotep/
(CRAG-6), 15 - 17 June 2016, University of Imhotep Mathematical Proceedings
Bamenda, Bamenda, Cameroon.
IBADAN UNIVERSITY LIBRARY
AFRICAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Imhotep Mathematical Proceedings
Volume 4, Numéro 1, (2017),
pp. 27 - 33.
Classifying a class of the fuzzy subgroups of the alternating groups An
M.E. Ogiugo
Department of Mathematics,
University of Ibadan, Ibadan, Abstract
Nigeria.
The aim of this paper is to classify the fuzzy subgroups of the alternating
ndjeyas@yahoo.fr group. First, an equivalence relation on *the set of all fuzzy subgroups of
a group G is defined. Without any equivalence relation on fuzzy subgroups
M. Enioluwafe of group G, the number of fuzzy subgroups is infinite, even for the trivial
group. Explicit formulae for the number of distinct fuzzy subgroup of finite
Department of Mathematics, alternating group are obtained in the particular case n = 5. Some inequalities
University of Ibadan, Ibadan, satisfied by this number are also established for n ≥ 5
Nigeria.
hervekalachi@gmail.com
Proceedings of the 6th annual workshop on
CRyptography, Algebra and Geometry http://imhotep-journal.org/index.php/imhotep/
(CRAG-6), 15 - 17 June 2016, University of Imhotep Mathematical Proceedings
Bamenda, Bamenda, Cameroon.
IBADAN UNIVERSITY LIBRARY
IMHOTEP - Math. Proc. 4 (2017), 27–33 IMHOTEP Mathematical
1608-9324/010027–26, DOI 13.1007/s00009-003-0000
©c 2017 Imhotep-Journal.org/Cameroon Proceedings
Classifying a class of the fuzzy subgroups of the
alternating groups An
M.E. Ogiugo and M. Enioluwafe
Abstract. The aim of this paper is to classify the fuzzy subgroups of the alternating group.
First, an equivalence relation on *the set of all fuzzy subgroups of a group G is defined.
Without any equivalence relation on fuzzy subgroups of group G, the number of fuzzy
subgroups is infinite, even for the trivial group. Explicit formulae for the number of distinct
fuzzy subgroup of finite alternating group are obtained in the particular case n = 5. Some
inequalities satisfied by this number are also established for n ≥ 5
Keywords. Fuzzy subgroups, chains of subgroups, maximal chains of subgroups,Alternating
groups, symmetric groups, recurrence relations.
I. Introduction
The concept of fuzzy sets was first introduced by Zadeh in 1965 (see[18]). The study of fuzzy
algebraic structures was started with the introduction of the concept of fuzzy subgroups by
Rosenfeld in 1971 (see[17]).The pioneering work of Zadeh on fuzzy subsets of a set and Rosenfeld
on fuzzy subgroups of a group led to the fuzzification of algebraic structures.
One of the most important problems of fuzzy group theory is to classify the fuzzy
subgroups of groups and to count all distinct fuzzy subgroups of finite groups. This topic has
enjoyed a rapid development in the last few years . In our case the corresponding equivalence
classes of fuzzy subgroups are closely connected to the chains of subgroups in G. The group
structures can be classified by assigning equivalence classes to its fuzzy subgroups. As a guiding
principle in determining the number of these classes, we first found the number of maximal
chains of G. Note that an essential role in solving our counting problem is played again by the
Inclusion-Exclusion Principle.
Sulaiman and Abd Ghafur [11] have counted the number of fuzzy subgroups of symmet-
ric group S2, S3 and alternating group A3 . Sulaiman[10] have constructed the fuzzy subgroups
of symmetric group S4 using the Maximal chain method, while Tarnauceanu [16] have also
computed the number of fuzzy subgroups of symmetric group S4 by the Inclusion -Exclusion
Principle.
The most familiar of the finite (non−abelian) simple groups are the alternating groups
An , which are subgroups of index 2 in the symmetric groups. The alternating group of degree
n is the only non- identity, proper normal subgroup of the symmetric group of degree n except
Communication presentée au 6ème atelier annuel sur la CRyptographie, Algèbre et Géométrie (CRAG-6), 15 - 17 Juin
2016, Université de Bamenda, Bamenda, Cameroun / Paper presented at the 6th annual workshop on CRyptography,
Algebra and Geometry (CRAG-6), 15 - 17 June 2016, University of Bamenda, Bamenda, Cameroon.
IBADAN UNIVERSITY LIBRARY
28 M.E. Ogiugo and M. Enioluwafe
when n = 1, 2, or 4. In cases n ≥ 2 , then the alternating group itself is the identity, but in the
case n = 4 , there is a second non- identity, proper, normal subgroup, the Klein four group.
The normal subgroups of the symmetric groups on infinite sets include both the corresponding
“alternating group” on the infinite set, as well as the subgroups indexed by infinite cardinals
whose elements fix all but a certain cardinality of elements of the set. For instance, the symmetric
group on a countably infinite set has a normal subgroup S consisting of all those permutations
which fix all but finitely many elements of the set. The elements of S are each contained
in a finite symmetric group, and so are either even or odd. The even elements of S form a
characteristic subgroup of S called the alternating group , and are the only other non- identity,
proper, normal subgroup of the symmetric group on a countably infinite set (see[2])
II. Preliminaries
Let G be a group with a multiplicative binary operation and identity e, and let µ : G→ [0, 1] be
a fuzzy subset of G. Then µ is said to be a fuzzy subgroup of G if (1) µ(xy) ≥ min{µ(x), µ(y)},
and (2) µ(x−1) ≥ µ(x) for all x, y ∈ G. The set {µ(x)|x ∈ G} is called the image of µ and is
denoted by µ(G) . For each α ∈ µ(G), the set µα : = {x ∈ G|µ(x) ≥ α} is called a level subset
of µ. It follows that µ is a fuzzy subgroup of G if and only if its level subsets are either empty
or subgroups of G. These subsets allow us to characterize the fuzzy subgroups of G (see [3]).
Two fuzzy subgroups µ and ν of G are equivalent, written as µ ∼ ν, if µ(x) ≥ µ(y)⇔
ν(x) ≥ ν(y) for all x, y ∈ G. It follows that µ ∼ ν if and only if µ and ν have the same set of
level Subgroups and two fuzzy subgroups µ, ν of G will be called distinct if µ ∼6 ν. (see[13]).
Hence there exits a one-to-one correspondence between the collection of the equivalence classes
of fuzzy subgroups of G and the collection of chains of subgroups of G which end in G. So,the
problem of counting all distinct fuzzy subgroups of G can be translated into a combinatorial
problem on the subgroup lattice L(G) of G .This notion of equivalence relation was in [10,13,14,
16] in order to enumerate fuzzy subgroups of certain families of finite groups. There is another
equivalence relation on the set of fuzzy subgroups used by Murali and Makamba [6, 7, 8, 9]
in order to enumerate fuzzy subgroups of certain families of finite abelian groups. Some other
different approaches to classify the fuzzy subgroups can be found in [ 4 ] and [ 5 ].
Most recent ,the problem of classifying the fuzzy subgroup of finite group G by using a new
equivalence relation ≈ on the lattice of all fuzzy subgroups of G, its definition has a consistent
group theoretical foundation, by involving the knowledge of the automorphism group associated
to G. The approach is motivated by the realization that in a theoretical study of fuzzy groups,
fuzzy subgroups are distinguished by their level subgroups and not by their images in [0, 1].
Consequently, the study of some equivalence relations between the chains of level subgroups of
fuzzy groups is very important. It can also lead to other significant results which are similar with
the analogous results in classical group theory (see [15]). In this paper we follow the notion of
the equivalence relation used in [15]. This equivalence relation generalizes that used in Murali’
s papers [6] - [9]. It is also closely connected to the concept of level subgroup.
One next goal is to describe the method that will be used in counting the chains of subgroups of
G. Let M1,M2, ...Mk be the maximal subgroups of G and denote by g(G) (respectively by h(G))
the number of maximal chain of subgroups in G (respectively the number of chains of subgroups
of G ending in G). The technique developed to obtain g(G) is founded on the following simple
remark: every maximal chain in G contains a unique maximal subgroup of G. In this way, g(G)
and g(Mi), i = 1, 2, . . . , k, are connected by the equality
∑k
g(G) = g(Mi) (1)
i=1
Imhotep Proc.
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Vol. 4 (2017) Classifying a class of the fuzzy subgroups of the alternating groups An 29
For finite cyclic groups , t(his equality leads to the)well- known formula
m1 +m2 + · · ·+ms (m1 +m2 + · · ·+ms)!g(Zn) = m1,m2, · · ·
= (2)
,ms m1!m2! · · ·ms!
In order to compute the number of all distinct fuzzy subgroups of a finite G which is
denoted by h(G), we shall apply the inclusion-Exclusion Principle. (see[15])∑k ∑ ⋂ kh(G) = 2 h(Mi)− h(Mi ∩Mi ) + ...+ (−1)k−1h( M i) (3)1 2
i=1 1≤i1<i2≤k i=1
III. Main Results
III.1. The cases n = 3, 4 and 5
Our problem is very simple for n = 3, since the Alternating group A3 is isomorphic to Cyclic
group of order 3. In the number of all distinct fuzzy subgroups of A3 is
h(A3) = h(C3) = 2 (4)
The alternating group A4 possesses five maximal subgroups, one isomorphic with D4 and four
isomorphic to C3. Therefore, we have
g(A4) = g(D4) + 4g(C3) = 7
The number of all distinct fuzzy subgroups of alternating group A4 was computed using the
Inclusion-Exclusion Principle (eqn3)
h(A4) = 24 (5)
In order to compute the number of all distinct fuzzy subgroups of alternating group A5, we
need to describe its maximal subgroups structure. It is well-known the simplicity of alternating
groups i.e., for n ≥ 5, the alternating group An is simple.
Definition[1]: A group G is simple if G has no normal subgroups other than {1} and G itself.
Such groups have remarkable properties.
By Lagrange’s Theorem, any group of prime order is simple. All other simple groups are non-
abelian. These small simple groups belong to two families An and PSL2(q) for n ≥ 5 and q ≥ 5
a prime power.The alternating group A5 exhibits exceptional isomorphisms:
A ∼5 = PSL2(4) ∼= PSL2(5)
The alternating group A5 possesses 21 maximal subgroups, of which 6 are isomorphic to D10,
10 are isomorphic to S3 and 5 are isomorphic to A4. So that we get:
∑21
g(A5) = g(Mi) = 10g(S3) + 6g(D10) + 5g(A4)
i=1
g(A5) = 10(4) + 6(8) + 5(7)
g(A5) = 123
Lemma 1. The number g(A5) of all the maximal chains of subgroups of the alternating group
A5 is 123
We deduce that a lower bound for g(An) , where n ≥ 5 is arbitrary.
Proposition 1. For n ≥ 5, the number g(An) of all maximal chains of subgroups of the al-
ternating group An satisfies the following inequality:
g(An) ≥ g(Sn−2) + ng(An−1) (6)
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30 M.E. Ogiugo and M. Enioluwafe
For instance the inequality (6) derived the lower bounds for n = 5, 6 as below:
g(A5) ≥ g(S3) + ng(A4)
g(A5) ≥ 4 + 5(7)
g(A5) ≥ 39
For n = 6
g(A6) ≥ g(S4) + ng(A5)
g(A6) ≥ 44 + 6(123)
g(A6) ≥ 782
Maximal Subgroup Structure of A5
Maximal Subgroups Generating sets Order Number
S3 〈(1, 2, 3), (1, 2)(4, 5)〉 6 10
A4 〈(1, 2, 3), (2, 3, 4)〉 12 5
D10 〈(2, 4)(3, 5), (1, 2, 3, 5, 4)〉 10 6
There exist 10 such maximal subgroups, all isomorphic to S3 :
1. , M1 = 〈(1, 2, 3), (1, 2)(4, 5)〉
2. M2 = 〈(1, 2, 4), (1, 2)(3, 5)〉,
3. M3 = 〈(1, 2, 5), (1, 2)(3, 4)〉,
4. M4 = 〈(1, 3, 4), (1, 3)(2, 5)〉,
5. M5 = 〈(1, 3, 5), (1, 3)(2, 4)〉,
6. M6 = 〈(1, 4, 5), (1, 4)(2, 3)〉,
7. M7 = 〈(2, 3, 4), (1, 5)(2, 3)〉,
8. M8 = 〈(2, 3, 5), (1, 4)(2, 3)〉,
9. M9 = 〈(2, 4, 5), (1, 3)(2, 4)〉,
10. M10 = 〈(3, 4, 5), (1, 2)(3, 4)〉,
There exist 5 such maximal subgroups, all isomorphic to A4 ;
1. M11 = 〈(1, 2, 3), (2, 3, 4)〉,
2. M12 = 〈(1, 2, 5), (1, 2, 3)〉,
3. M13 = 〈(1, 4, 5), (1, 2, 5)〉,
4. M14 = 〈(3, 4, 5), (1, 4, 5)〉,
5. M15 = 〈(2, 3, 4), (3, 4, 5)〉,
There exist 6 such maximal subgroups, all isomorphic to D10 ;
1. M16 = 〈(2, 4)(3, 5), (1, 2, 3, 5, 4)〉,
2. M17 = 〈(2, 5)(3, 4), (1, 2, 4, 3, 5)〉,
3. M18 = 〈(2, 3)(4, 5), (1, 2, 5, 4, 3)〉,
4. M19 = 〈(2, 4)(3, 5), (1, 3, 2, 4, 5)〉,
5. M20 = 〈(2, 5)(3, 4), (1, 3, 5, 2, 4)〉,
6. M21 = 〈(2, 3)(4, 5), (1, 4, 3, 2, 5)〉,
IV. Counting the number of fuzzy subgroups of alternating group A5
The problem of counting all distinct fuzzy subgroups of G can be translated into combinatorial
problem on the subgroup lattice L(G) of G: finding the number of all chain of subgroups of G
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Vol. 4 (2017) Classifying a class of the fuzzy subgroups of the alternating groups An 31
that terminates in G. Clearly, we obtain that in any group with at least two elements there are
more distinct fuzzy subgroup than subgroups.
Since the maximal subgroups Mi of A5 and their intersections have been precisely determined
with the help of the computational group theory system GAP.
This has been used in[12, 15] to obtain explicit formulas of h(D2n) for some classes of
positive integers n. Recall here that
2m1 ( )
h(D ) = Pm1+12n 1 + p− 1
− 2
p1 1
if
n = pm11
and, in particular , h(D4) = 8, h(D6) ∼= h(S3) = 10 , h(D8) = 32 , h(D10) = 68
h(C × C ) ∼= h(C ) ∼2 2 4 = h(D4) = 8
h(A ∼3) = h(C3) = 2.
It follows that :
Mi ∩Mi ∩ · · · ∩Mi = {e}, fo∑r all r ≥ 8 and all 0 ≤ i1 < i2 < · · · < ir ≤ 211 2 r
c = (−1)r−1r h(Mi ∩Mi ∩ · · · ∩M1 2 i )r
0≤i1<i2···<ir≤21
We have
c1 = ((10h(S)3)) + 6h(D10) + 5h(A4)) = 304c2 = −(30h(Ae) + 30h(C3) + 150h(C2)) = −390
21
c3 = ( )− 160 + 10h(C3) + 150h(C2) = 14903
− 21c4 = − 75 + 75h(C2) = −6060
( ) 4 ( )
( 21c5 = )− 2115 + 15h(C2) = 20364 c6 = −( ) = −542645 6
21 21
c7 = ( ) = 116280 c8 = − ( )= −2034907 8
21 21
c9 = ( )= 293930 c10 = −( ) = −3527169 10
( 21 21c11 = = 352716 c = − = −29393011 ) 12 ( 12 )
( 21 ) 21c13 = = 203490 c14 = −( ) = −11628013 14
( 21 21c15 = ) = 54264 c16 = −( ) = −2034915 16
( 21 ) 21c17 = = 5785 c18 = −( ) = −133017 18
21 21
c19 = ( ) = 210 c20 = − = −2119 20
21
c21 = = 121∑21
h(A5) = 2 cr = 408
r=1
Theorem 4 The number h(A5) of all distinct fuzzy subgroups of the alternating group A5 is 408
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32 M.E. Ogiugo and M. Enioluwafe
V. An Upper Bound for h(An), n ≥ 5
Finally, for the cases of n = 3, 4, 5 considered, the number of distinct fuzzy subgroups of the
alternating group was computed by above-mentioned method (Inclusion-Exclusion Principle)
and group Isomorphism for the case of n = 3. The alternating group of degree n is very difficult
to describe all its maximal subgroups when n becomes large and so the method of computing
the number of all distinct fuzzy subgroups will be tedious . It necessitate develop a lower bound
and upper bound for the number of all distinct fuzzy subgroups for larger classes of n. In order
of estimate (3), the group isomorphism for the maximal subgroups and their direct computation
of intersections.
Studying the classification of certain maximal subgroups of the alternating groups, we can easily
see the only case where Am has a maximal subgroup that is isomorphic to a symmetric group
Sn that acts transitively but imprimitively on {1, . . . ,m} is when (n,m) = (4, 6) . Also, Am
has a maximal subgroup that is isomorphic to An , which does act transitively on {1, . . . ,m}
that is (n,m) = (5, 6). The subgroup structure of An and Sn is described by the O’Nan-Scott
Theorem. It happens that classification of maximal subgroups is convenient in terms of concept
of transitivity (see [1]). Since our method (Inclusion and Exclusion Principle) of the computing
the number of fuzzy subgroups of An is based on the direct calculation of intersection of their
maximal subgroups. This allows us to deduce the following inequalities :
Theorem 5 . For n ≥ 5, the upper bound for the number h(An) of all distinct fuzzy subgroups
of the alternating grou(p∑An satisfi(es the fo)llowing inequali∑ty:n−2 n−2( )n n
h(An) ≤ 2 (−1)r h(A ) + h(Sr + 2 n−r−1 r + 2 n−r−2)
r=0
n∑−2( )
r=)0
n
+ h(D
r + 2 2r+4
) (7)
r=0
For example n = 5, then we have that (7) becomes
h(A5) ≤ 592 + 10h(A4)
which implies that
h(A5) < 602 (8)
For n = 6, and a upper bound for h(A6) becomes
h(A6) ≤ 6468 + 15h(A5)
which leads to
h(A6) < 6483 (9)
VI. Conclusion
The study concerning the classification of the fuzzy subgroups of (finite) groups is a significant
aspect of fuzzy group theory. The problem of counting the number of distinct fuzzy subgroups
relative to the the notion of the equivalence relation. Without any equivalence relation on
fuzzy subgroups of finite group, the number of fuzzy subgroups is infinite, even for the trivial
group. These equivalence relations provide settings for classifying the fuzzy subgroups of finite
groups. The group structures can be classified by the assigning equivalence classes to its fuzzy
subgroups. This will surely constitute the subject of further research on the classification of the
fuzzy subgroups of finite symmetric groups.
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Vol. 4 (2017) Classifying a class of the fuzzy subgroups of the alternating groups An 33
VI.1. Further Research
• Establishing some explicit formulas for g(Sn) and h(Sn) for other n ≥ 5
• Establishing some explicit formulas for g(An) and h(An) for other n ≥ 5
• Develop a method to find some upper bounds for g(Sn) and h(Sn), n ≥ 5
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M.E. Ogiugo
e-mail: ekpenogiugo@gmail.com
Department of Mathematics, University of Ibadan, Ibadan, Nigeria.
M. Enioluwafe
e-mail: michael.enioluwafe@gmail.com
Department of Mathematics, University of Ibadan, Ibadan, Nigeria.
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