EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5889
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Jordan-Hölder Theorem for Multigroups

Paul Augustine Ejegwa1, Nasreen Kausar2, Musa Adeku Ibrahim3,
Tonguc Cagin4,∗

1 Department of Mathematics, Joseph Sarwuan Tarka University, P.M.B. 2373, Makurdi,
Nigeria
2 Department of Mathematics, Faculty of Arts and Sciences, Balikesir University, 10145
Balikesir, Türkiye
3 Department of Mathematics, Federal University Lokoja, Kogi State, Nigeria
4 College of Business Administration, American University of the Middle East, Kuwait

Abstract. Multigroup theory is the application of multisets to the theory of groups. Many group’s
theoretic notions have been studied in multigroup theory, however, the ideas of maximal normal
subgroup, simple group, normal series, composition series, and the Jordan-Hölder Theorem are
yet to be investigated in multiset context. In this article, we define simple multigroup, maximal
normal submultigroup, normal series for multigroup, and composition series for multigroup with
examples. With these concepts, we establish the Jordan-Hölder Theorem in multigroup theory.
It is shown that every finite multigroup defined over a finite group has a composition series. In
addition, it is established that every finite multigroup defined over a finite group has at least two
composition series which are equivalent.

2020 Mathematics Subject Classifications: 03E72, 06D72, 11E57, 19A22

Key Words and Phrases: Multiset, Multigroup, Order of multigroup, Simple multigroup, Max-
imal normal submultigroup, Normal series, Composition series

One constraint of set theory is the refusal to allow repeated elements in a collection,
which is admissible in real-world applications. The word ”multiset” refers to an extensional
set where an element can be repeated in a collection [1]. According to DeBruijin [2], the
concept of multiset was introduced to D. E. Knuth by N. G. de Bruijn in a private message,
and since then, the word has been used to depict a set with repeated elements/members.
The relevance of multiset has led to many studies and applications in a number of fields
[3–9]. By relaxing the condition of definite collection in set, Zadeh [10] introduced fuzzy
sets, which was applied to group theory by proposing fuzzy group theory [11]. Some
properties of the fuzzy group theory were discussed [12–16]. Nazmul et al. [17] utilized
multisets in group theory to introduce the theory of multigroups.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5889

Email addresses: ejegwa.augustine@uam.edu.ng (P. A. Ejegwa), nasreen.kausar@balikesir.edu.tr
(N. Kausar), adekubash@gmail.com (M. A. Ibrahim), tonguc.cagin@aum.edu.kw (T. Cagin)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 2 of 13

A comprehensive research on multigroup theory was conducted in [18] and some re-
sults in multigroup theory were discussed [19, 20]. The concepts of highly invariant
submultigroups, characteristic submultigroups, normal submultigroups, and Frattini sub-
multigroups were investigated in multigroup settings, yielding some relevant results [21–
24]. In addition, the order of multigroups, cyclic multigroups, comultisets, and factor
multigroups were established [25–28]. The studies in [29–34] examined various notions in
multigroup contexts, such as direct products and actions. Some algebraic systems have
been examined using the idea of multisets [35–38]. Finally, the idea of soluble multigroups
was introduced and many of its properties were studied in [39].

Although many group theoretic concepts have been addressed under multiset context,
the notions of maximal normal subgroup, simple group, normal series, composition series,
and the Jordan-Hölder Theorem are yet to be studied in multiset domain. Hence, it
is appropriate to investigate simple multigroup, maximal normal submultigroup, normal
series for multigroup, composition series for multigroup, and the Jordan-Hölder Theorem
for multigroups because the necessary concepts needed for the establishment of these
concepts have been studied in multigroup theory. Thus, this article establishes simple
multigroup, maximal normal submultigroup, normal series for multigroup, composition
series for multigroup, and the Jordan-Hölder Theorem for multigroups, respectively. The
remainder of the article is organized as follows: Section 2 presents the preliminaries for
the study, Section 3 covers the main results of the articles, and Section 4 concludes and
makes suggestions for further research.

1. Preliminaries

Let S and G represent a non-empty set and a group, respectively.

Definition 1 ([10]). A fuzzy subset F of S is presented as:

F = {⟨s,Fm(s)⟩ | s ∈ S}, (1)

where Fm : S → [0, 1] is the membership degree of s ∈ S.

Definition 2 ([11]). A fuzzy subset F of G is a fuzzy subgroup of G if

(i) Fm(xy) ≥ min
{
Fm(x),Fm(y)

}
∀ x, y ∈ G,

(ii) Fm(x−1) = Fm(x) ∀ x ∈ G.

In addition, Fm(e) = Fm(xx−1) ≥ min
{
Fm(x),Fm(x)

}
= Fm(x) ∀ x ∈ G, where e is the

unit element of G.

Definition 3 ([6]). A multiset D of S is a pair ⟨S,CD⟩, where

CD : S → N = {1, 2, ...} (2)

is a function, such that for s ∈ S implies D(s) = CD(s) > 0 and CD(s) is the multiplicity
of s in D. If CD(s) = 0, then s /∈ S.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 3 of 13

Definition 4 ([8]). Suppose D and E are multisets of S, then

(i) D = E ⇐⇒ CD(s) = CE(s) ∀ s ∈ S,

(ii) D ⊆ E ⇐⇒ CD(s) ≤ CE(s) ∀ s ∈ S,

(iii) D ∩ E =⇒ CD∩E(s) = min{CD(s), CE(s)} ∀ s ∈ S,

(iv) D ∪ E =⇒ CD∪E(s) = max{CD(s), CE(s)} ∀ s ∈ S,

(v) D⊕ E =⇒ CD⊕E(s) = CD(s)⊕ CE(s) ∀ s ∈ S.

Definition 5 ([17]). A multiset D of G is called a multigroup of G if:

(i) CD(xy) ≥ min{CD(x), CD(y)} ∀ x, y ∈ G,

(ii) CD(x
−1) = CD(x) ∀ x ∈ G.

It is worthy to note that, CD(e) ≥ CD(x) ∀ x ∈ X since

CD(e) = CD(xx
−1) ≥ min{CD(x), CD(x)} = CD(x) ∀x ∈ G.

In addition, D∗ defined by D∗ = {x ∈ G | CD(x) > 0} is a subgroup of G.

Definition 6 ([26]). If D is a multigroup of G, then the order of D is the sum of the
multiplicities for each of the elements in D. It is mathematically presented as:

|D| =
n∑

i=1

CD(xi) ∀xi ∈ G. (3)

Definition 7 ([31]). A multigroup D of G is commutative if CD(xy) = CD(yx) ∀ x, y ∈ G.
If G is commutative, then a multigroup D of G is a commutative multigroup.

Definition 8 ([31]). Suppose D and E are multigroups of G, then D is a submultigroup
of E if D ⊆ E. Again, D is a proper submultigroup of E if D ⊆ E and D ̸= E.

Definition 9 ([22]). Suppose D is a submultigroup of a multigroup E of G, then D is
normal in E denoted by D ◁ E if CD(xy) = CD(yx) ⇐⇒ CD(y) = CD(x

−1yx) ∀ x, y ∈ G.
Certainly, any normal submultigroup is commutative and self-normal.

Definition 10 ([28]). Let D be a submultigroup of a multigroup E of G. Then, the
submultiset yD of E for y ∈ G defined by CyD(x) = CD(y

−1x) ∀ x ∈ G is a left comultiset
of D. Similarly, Dy of E such that CDy(x) = CD(xy

−1) ∀ x ∈ G is a right comultiset of
D.

Definition 11 ([18]). Suppose D and E are multigroups of G. Then, the product D ◦ E
is a multiset of G defined as follows:

CD◦E(x) =

{ ∨
x=yz min{CD(y), CE(z)}, if ∃ y, z ∈ G where x = yz

0, otherwise.
(4)

Definition 12 ([28]). Suppose E is a multigroup of G and D a normal submultigroup in E.
Then, the set of right/left comultisets of D such that CxD◦yD(z) = CxyD(z) ∀ x, y, z ∈ G
is a factor/quotient multigroup of E by D, represented as E/D.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 4 of 13

2. Main Results

Before the introduction of normal series, composition series, and Jordan-Hölder therorem
under multigroups, we first reiterate solvable multigroups as established in [39] as follows:

Definition 13. For every finite multigroup D of a finite G, there is a chain of consecutive
submultigroups of D:

D0 ⊆ D1 ⊆ · · · ⊆ Dn = D, (5)

where (D0)∗ = (D1)∗ = · · · = (Dn)∗ = D∗.

The chain of the consecutive submultigroups is also presented as:

CD0(x) ≤ CD1(x) ≤ · · · ≤ CDn(x) = CD(x), ∀x ∈ G (6)

such that (D0)∗ = (D1)∗ = · · · = (Dn)∗ = D∗.

Definition 14. If D is a multigroup of G, then D is solvable if it has a chain of consecutive
submultigroups:

D0 ⊆ D1 ⊆ · · · ⊆ Dn = D (7)

such that (D0)∗ = (D1)∗ = · · · = (Dn)∗ = D∗, where Di−1 ◁Di and Di/Di−1 is commuta-
tive ∀ 1 ≤ i ≤ n.

The finite chain of consecutive submultigroups of D is a solvable series for D denoted
by Di. In fact, the solvable series for D is presented as:

D0 ◁D1 ◁ · · · ◁Dn = D. (8)

Next, we shall define the concepts of maximal normal submultigroup of a multigroup
and simple multigroup. From the concept of normal submultigroup in Definition 9, we
define a maximal normal submultigroup of a multigroup as follows:

Definition 15. Let C and D be multigroups of G such that C ◁D. Then

(i) C is a maximal non-trivial normal submultigroup if it is the largest proper non-trivial
normal submultigroup of D.

(ii) D is simple if it has no proper non-trivial normal submultigroup.

Remark 1. A submultigroup B of a multigroup D of G is trivial if:

(i) B is the identity element or the identity element with multiplicity,

(ii) B is a subgroup of G or G itself,

(iii) B is the same as D.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 5 of 13

Example 1. Suppose G = {1, d, d2, d3} where d4 = 1, d−1 = d3, (d2)−1 = d2, and
(d3)−1 = d. Then, a multigroup of G is:

D = {1, 1, 1, 1, d, d, d2, d2, d2, d3, d3}.

Certainly, D is commutative. The submultigroups of D can be presented in terms of:

(i) the properties of G without multiplicity (since every subgroup is a submultigroup of
trivial multiplicity),

(ii) D∗ with multiplicity,

(iii) the properties of G and multiplicity.

By Case (i), the submultigroups of D are as follows:

D0 = {1},D1 = {1, d2},D2 = D∗ = {1, d, d2, d3}.

In this case, D is not included as a submultigroup of itself because it contains multiplicity.
Clearly, {1, d, d3} is not a submultigroup of D because d.d, d3.d3 /∈ {1, d, d3}. Since D0,
D1, and D2 are trivial submultigroups of D (because their multiplicity is 1), D has neither
proper non-trivial normal submultigroup nor a maximal non-trivial normal submultigroup
and hence, D is a simple multigroup.

Using Case (ii), the submultigroups of D are in Table 1:

Table 1: Submultigroups of D based on Case (ii)

Submultigroups and their structures

D̂1 = D2 = {1, d, d2, d3}, D̂2 = {1, 1, d, d2, d3},
D̂3 = {1, 1, d, d, d2, d2, d3, d3},

D̂4 = {1, 1, 1, d, d2, d3}
D̂5 = {1, 1, 1, d, d, d2, d2, d3, d3},

D̂6 = {1, 1, 1, d, d, d2, d2, d2, d3, d3},
D̂7 = {1, 1, 1, 1, d, d2, d3},

D̂8 = {1, 1, 1, 1, d, d, d2, d2, d3, d3},
D̂9 = D = {1, 1, 1, 1, d, d, d2, d2, d2, d3, d3}

Among the list, D̂1 and D̂9 are trivial submultigroups because D̂1 = D∗ and D̂9 is a
submultigroup of itself, D̂2–D̂8 are proper non-trivial normal submultigroups of D, and
D̂8 is the maximal non-trivial normal submultigroup of D. Hence, D is not simple.

By Case (iii), the submultigroups of D are in Table 2:



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 6 of 13

Table 2: Submultigroups of D based on Case (iii)

Submultigroups and their structures

D0 = {1}, D1 = {1, d2}, D2 = D̂1 = {1, d, d2, d3},
D̂2 = {1, 1, d, d2, d3}, D̂3 = {1, 1, d, d, d2, d2, d3, d3},

D̂4 = {1, 1, 1, d, d2, d3}, D̂5 = {1, 1, 1, d, d, d2, d2, d3, d3},
D̂6 = {1, 1, 1, d, d, d2, d2, d2, d3, d3}, D̂7 = {1, 1, 1, 1, d, d2, d3},

D̂8 = {1, 1, 1, 1, d, d, d2, d2, d3, d3},
D̂9 = D = {1, 1, 1, 1, d, d, d2, d2, d2, d3, d3},

Ḋ1 = {1, 1}, Ḋ2 = {1, 1, 1}, Ḋ3 = {1, 1, 1, 1}, Ḋ4 = {1, 1, d2},
Ḋ5 = {1, 1, d2, d2}, Ḋ6 = {1, 1, 1, d2}, Ḋ7 = {1, 1, 1, d2, d2},

Ḋ8 = {1, 1, 1, d2, d2, d2}, Ḋ9 = {1, 1, 1, 1, d2},
Ḋ10 = {1, 1, 1, 1, d2, d2}, Ḋ11 = {1, 1, 1, 1, d2, d2, d2}

Among the list, D0–D2, Ḋ1–Ḋ3, and D̂9 are trivial submultigroups of D. Submulti-
groups D̂2–D̂8 and Ḋ4–Ḋ11 are proper non-trivial normal submultigroups of D, and D̂8 is
the maximal non-trivial normal submultigroups ofD. Again, D is not a simple multigroup.

Example 2. In a symmetry group Sn for n = 3 (i.e., S = {1, 2, 3}), A3 = {ρ0, ρ1, ρ2} ⊆ S3

is a simple group (ρ−1
1 = ρ2, ρ−1

2 = ρ1, and ρ0 is the identity element). Using A3, a
multigroup defined over A3 is:

E = {ρ0, ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}.

Certainly, A3 is a trivial multigroup. Since ρ1.ρ2 = ρ2.ρ1 = ρ0, E is commutative.
By Case (i), the submultigroups of E are as follows:

E0 = {ρ0}, E1 = A3 = E∗ = {ρ0, ρ1, ρ2}.

Because E0 and E1 are trivial, then E is a simple multigroup (because it has no proper non-
trivial normal submultigroup and hence, no maximal non-trivial normal submultigroup).

By Case (ii), the submultigroups of E are in Table 3:

Table 3: Submultigroups of E based on Case (ii)

Submultigroups and their Structures

Ê1 = E1 = {ρ0, ρ1, ρ2}, Ê2 = {ρ0, ρ0, ρ1, ρ2},
Ê3 = {ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}, Ê4 = {ρ0, ρ0, ρ0, ρ1, ρ2},

Ê5 = E = {ρ0, ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}

Here, it is observed that Ê1 and Ê5 are trivial, and Ê2–Ê4 are proper non-trivial normal
submultigroups of E without a maximal non-trivial normal submultigroup ofD since either
Ê3 ⊈ Ê4 or Ê4 ⊈ Ê3. Hence, E is not simple.

By Case (iii), the submultigroups of E are in Table 4:



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 7 of 13

Table 4: Submultigroups of E based on Case (iii)

Submultigroups and their structures

E0 = {ρ0}, Ê1 = E1 = {ρ0, ρ1, ρ2}, Ê2 = {ρ0, ρ0, ρ1, ρ2},
Ê3 = {ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}, Ê4 = {ρ0, ρ0, ρ0, ρ1, ρ2},
Ê5 = E = {ρ0, ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}, Ė1 = {ρ0, ρ0},

Ė2 = {ρ0, ρ0, ρ0}

Here, E0, Ê1, Ė1, Ė2 and Ê5 are trivial, and Ê2–Ê4 are proper non-trivial normal
submultigroups of E without a maximal non-trivial normal submultigroup of E since either
Ê3 ⊈ Ê4 or Ê4 ⊈ Ê3. Hence, E is not a simple multigroup.

Example 3. Let G = {1, a, a2, a3, b, ab, a2b, a3b} be a group of order 8 with two generators
a and b, which satisfy the relations:

(i) a4 = b2 = 1 and ba = a3b = a−1b, which is a group recognize as D4.

(ii) a4 = 1, a2 = b2, and ba = a3b, which is a group of unit quaternions.

Indeed, a−1 = a3, (a3)−1 = a, (a2)−1 = a2, b−1 = b, (ab)−1 = a3b, (a2b)−1 = a2b, and
(a3b)−1 = ab. Certainly, G is non-abelian since b(ab) ̸= (ab)b because b(ab) = (ba)b =
a3b2 = a3 and (ab)b = ab2 = a.

Using G, a multigroup defined over G is:

F = {1, 1, 1, a, a, a2, a2, a2, a3, a3, b, b, ab, ab, a2b, a2b, a3b, a3b},

which is also non-commutative.
The following structures in Table 5 are the submultigroups of F:

Table 5: Submultigroups of F

Submultigroups and their structures

F0 = {1}, F1 = {1, 1}, F2 = {1, 1, 1}, F3 = {1, a, a2, a3},
F4 = {1, 1, a, a2, a3}, F5 = {1, 1, a, a, a2, a2, a3, a3},

F6 = {1, 1, 1, a, a2, a3}, F7 = {1, 1, 1, a, a, a2, a2, a3, a3},
F8 = {1, 1, 1, a, a, a2, a2, a2, a3, a3}, F9 = {1, a2},

F10 = {1, 1, a2, a2}, F11 = {1, 1, 1, a2, a2, a2}, F12 = {1, b},
F13 = {1, 1, b}, F14 = {1, 1, b, b}, F15 = {1, 1, 1, b, b},

F16 = {1, a, a2, a3, b, ab, a2b, a3b},
F17 = {1, 1, a, a2, a3, b, ab, a2b, a3b},

F18 = {1, 1, a, a, a2, a2, a3, a3, b, b, ab, ab, a2b, a2b, a3b, a3b},
F19 = {1, 1, 1, a, a2, a3, b, ab, a2b, a3b},

F20 = {1, 1, 1, a, a, a2, a2, a3, a3, b, b, ab, ab, a2b, a2b, a3b, a3b},
F = {1, 1, 1, a, a, a2, a2, a2, a3, a3, b, b, ab, ab, a2b, a2b, a3b, a3b}



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 8 of 13

Here, F0–F3, F9, F12, and F are trivial. The submultigroups F4–F8, F10, F11, and
F13–F20 are proper non-trivial normal submultigroups of F. Hence, F is not a simple
multigroup. Again, the maximal non-trivial normal submultigroup of F is F20.

Remark 2. Every multigroup whose submultigroups are trivial is a simple multigroup.

To see this, given a multigroup E = {ρ0, ρ0, ρ1, ρ2} defined over A3 = {ρ0, ρ1, ρ2}.
Then, the submultigroups of E are:

E0 = {ρ0},
E1 = {ρ0, ρ1, ρ2},

E2 = {ρ0, ρ0, ρ1, ρ2} = E.

Among these submultigroups of E, we notice that all of them are trivial. Thus, E is
simple.

Definition 16. Let G be a finite group and D be a multigroup of G with a finite multi-
plicity. Then, D has a normal series if there exist:

CD0(x) ≤ CD1(x) ≤ · · · ≤ CDn(x) = CD(x) ∀x ∈ G, (9)

such that (D0)∗ = (D1)∗ = · · · = (Dn)∗ = D∗ and Di ◁Di+1 ∀ 0 ≤ i ≤ n− 1.

Example 4. Using the submultigroups of D in Example 1, we observe that normal series
only exists for Case (ii) and Case (iii). It does not exist for Case (i) because (D0)∗ ̸=
D∗, (D1)∗ ̸= D∗, and (D2)∗ ̸= D∗. The normal series for Case (ii) is identical to Case
(iii) because all the submultigroups in Case (ii) are in Case (iii), and the rest of the
submultigroups in Case (iii) do not share the same elements as G. Thus, the normal series
for D are:

D̂1 ⊆ D̂2 ⊆ D̂4 ⊆ D̂7 ⊆ D̂8 ⊆ D̂9 = D,

D̂1 ⊆ D̂2 ⊆ D̂3 ⊆ D̂5 ⊆ D̂6 ⊆ D̂8 ⊆ D̂9 = D,

D̂1 ⊆ D̂2 ⊆ D̂4 ⊆ D̂5 ⊆ D̂6 ⊆ D̂8 ⊆ D̂9 = D.

Certainly, D̂i ◁ D̂i+1 ∀ 0 ≤ i ≤ n− 1.

Example 5. Using the submultigroups of E in Example 2, we have the following normal
series:

Ê1 ⊆ Ê2 ⊆ Ê3 ⊆ Ê5 = E,

Ê1 ⊆ Ê2 ⊆ Ê4 ⊆ Ê5 = E,

where Êi ◁ Êi+1 ∀ 0 ≤ i ≤ n− 1.

Closely related to normal series is the concept of composition series.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 9 of 13

Definition 17. Let G be a finite group and D be a multigroup of G with a finite multiplic-
ity. Then, D has a composition series if there exist a chain of consecutive submultigroups:

CD0(x) ≤ CD1(x) ≤ · · · ≤ CDn(x) = CD(x) ∀x ∈ G, (10)

such that (D0)∗ = (D1)∗ = · · · = (Dn)∗ = D∗ with the properties

(i) Di ◁Di+1 ∀ 0 ≤ i ≤ n− 1,

(ii) Di+1/Di is simple ∀ 0 ≤ i ≤ n− 1.

For Example 1, the composition series for D are:

D̂1 ⊆ D̂2 ⊆ D̂4 ⊆ D̂7 ⊆ D̂8 ⊆ D̂9 = D,

D̂1 ⊆ D̂2 ⊆ D̂3 ⊆ D̂5 ⊆ D̂6 ⊆ D̂8 ⊆ D̂9 = D,

D̂1 ⊆ D̂2 ⊆ D̂4 ⊆ D̂5 ⊆ D̂6 ⊆ D̂8 ⊆ D̂9 = D.

Certainly, D̂i ◁ D̂i+1 ∀ 0 ≤ i ≤ n− 1 and D̂i+1/D̂i is simple ∀ 0 ≤ i ≤ n− 1.
For Example 2, we have the following normal series:

Ê1 ⊆ Ê2 ⊆ Ê3 ⊆ Ê5 = E,

Ê1 ⊆ Ê2 ⊆ Ê4 ⊆ Ê5 = E.

Since Êi for i = 1, 2, 3, 4, 5 are normal, then Êi ◁ Êi+1 ∀ 0 ≤ i ≤ n − 1 and Êi+1/Êi is
simple ∀ 0 ≤ i ≤ n− 1.

Example 6. Let C = {0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5} be a multigroup of Z6. Then, the
non-trivial submultigroups of C which share the same elements as G are:

Ĉ1 = {0, 1, 2, 3, 4, 5},
Ĉ2 = {0, 0, 1, 2, 3, 4, 5},

Ĉ3 = {0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5},
Ĉ4 = {0, 0, 0, 1, 2, 3, 4, 5},

Ĉ5 = {0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5} = C.

Then, the composition series are:

Ĉ1 ⊆ Ĉ2 ⊆ Ĉ3 ⊆ Ĉ5 = C,

Ĉ1 ⊆ Ĉ2 ⊆ Ĉ4 ⊆ Ĉ5 = C.

Because Ĉi ◁ C for i = 1, 2, 3, 4, then Ĉi ◁ Ĉi+1 ∀ 0 ≤ i ≤ n − 1 and Ĉi+1/Ĉi is simple ∀
0 ≤ i ≤ n− 1.

Theorem 1. Every finite multigroup defined over a finite group has a composition series.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 10 of 13

Proof. Let D be a finite multigroup over a finite G. We establish the proof by the
principle of induction. Suppose every finite multigroup of order less than |D| has a compo-
sition series. Now, if D is simple, then there is no composition series since no non-trivial
proper normal submultigroup exist. On the other hand, if D is not simple, then there
must be a non-trivial proper normal submultigroup. Since D is finite, there is a maximal
non-trivial normal submultigroup in D, which we denote as B. Certainly, |B| < |D|.

By induction, |B| has a composition series:

CB0(x) ≤ CB1(x) ≤ · · · ≤ CBn(x) = CB(x) ∀x ∈ G.

But then, B ◁D since B is maximal in D, and so

CB0(x) ≤ CB1(x) ≤ · · · ≤ CBn(x) = CB(x) ≤ CD(x) ∀x ∈ G,

which is the composition series for D.

Theorem 2 (The Jordan-Hölder Theorem). Every finite multigroup defined over a finite
group has at least two composition series which are equivalent.

Proof. Let D be a finite multigroup over a finite group G. Suppose we have two
composition series for D:

CD0(x) ≤ CD1(x) ≤ · · · ≤ CDn(x) = CD(x) ∀x ∈ G,

CB0(x) ≤ CB1(x) ≤ · · · ≤ CBm(x) = CD(x) ∀x ∈ G,

such that Di ◁Di+1 with Di+1/Di simple ∀ 0 ≤ i ≤ n− 1 and Bj ◁Bj+1 with Bj+1/Bj

simple ∀ 0 ≤ j ≤ m−1. We need to prove that n = m and (D1/D0,D2/D1, · · · ,Dn/Dn−1)
is a rearrangement (denoted as ∼) of (B1/B0,B2/B1, · · · ,Bm/Bm−1).

We prove by induction on |D|. For |D| = 1, the result is trivial. AssumeDn−1 = Bm−1,
the result follows by induction. Then, assume Dn−1 ̸= Bm−1. Set Φ = Dn−1, Ψ = Bm−1,
and Ω = Φ ∩Ψ, where Ω is a maximal submultigroup of Dn−1 and Bm−1.

Now, Ω has a composition series, CΩ0(x) ≤ CΩ1(x) ≤ · · · ≤ CΩt(x) = CΩ(x) ∀x ∈ G.
Then,

CD0(x) ≤ CD1(x) ≤ · · · ≤ CDn−1(x) = CΦ(x) ∀x ∈ G and

CΩ0(x) ≤ CΩ1(x) ≤ · · · ≤ CΩt(x) = CΩ(x) ≤ CΦ(x) ∀x ∈ G

are both composition series for Φ. By induction, we have n− 1 = t+ 1 ⇒ n− 2 = t, and

(D1/D0,D2/D1, · · · ,Dn−1/Dn−2) ∼ (Ω1/Ω0,Ω2/Ω1, · · · ,Ωt/Ωt−1,Φ/Ω). (11)

Similarly,

CB0(x) ≤ CB1(x) ≤ · · · ≤ CBm−1(x) = CΨ(x) ∀x ∈ G and

CΩ0(x) ≤ CΩ1(x) ≤ · · · ≤ CΩt(x) = CΩ(x) ≤ CΨ(x) ∀x ∈ G



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 11 of 13

are both composition series for Ψ. Thus, m− 1 = t+ 1 ⇒ m− 2 = t, and

(B1/B0,B2/B1, · · · ,Bm−1/Bm−2) ∼ (Ω1/Ω0,Ω2/Ω1, · · · ,Ωt/Ωt−1,Ψ/Ω). (12)

From m − 1 = t + 1 and n − 1 = t + 1, we have n = m. By appending D/Φ to both
sides of (11), we have

(D1/D0, · · · ,Dn−1/Dn−2,D/Dn−1) ∼ (Ω1/Ω0, · · · ,Ωt/Ωt−1,Φ/Ω,D/Φ). (13)

Similarly, appending D/Ψ to both sides of (12), we have

(B1/B0, · · · ,Bm−1/Bm−2,D/Bm−1) ∼ (Ω1/Ω0, · · · ,Ωt/Ωt−1,Ψ/Ω,D/Ψ). (14)

The right hand side of (13) and (14) are identical except (Φ/Ω,D/Φ) and (Ψ/Ω,D/Ψ).
Hence, (Φ/Ω,D/Φ) ∼ (Ψ/Ω,D/Ψ) and so

(D1/D0, · · · ,Dn/Dn−1) ∼ (B1/B0, · · · ,Bm/Bm−1).

3. Conclusion

In this paper, the notions of simple multigroup, maximal normal submultigroup, nor-
mal series for multigroup, and composition series for multigroup were defined as algebraic
structures in multiset context and characterized with examples and some results. In addi-
tion, it was proven that every finite multigroup defined over a finite group has a compo-
sition series. Finally, the Jordan-Hölder Theorem was presented in multiset context, and
it was established that every finite multigroup defined over a finite group has at least two
composition series which are equivalent. The multigroup structures can be applicable in
cryptography for the security of data, molecular symmetry and molecular behavior during
chemical reactions, quantum mechanics and particle physics, algorithms in areas related
to coding theory and data structure, robotics and computer graphics, etc. For further
investigation, the concept of nilpotency in multigroup setting could be considered.

Acknowledgements

We greatly appreciate the editor-in-chiefs and the reviewers for their valuable com-
ments, which have improved the quality of the work.

References

[1] D Knuth. The art of computer programming. Semi Numerical Algorithms, Second
Edition, volume 2. Addison-Wesley, Reading, Massachusetts, 1981.

[2] N G DeBruijin. Denumerations of rooted trees and multisets. Discrete Appl. Math.,
6:25–33, 1983.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 12 of 13

[3] W D Blizard. Multiset theory. Notre Dame J. logic, 31:36–65, 1989.
[4] W D Blizard. The development of multiset theory. Modern Logic, 1(4):319–352, 1991.
[5] K P Girish and S J John. Relations and functions in multiset context. Inf. Sci.,

179:758–768, 2009.
[6] S P Jena, S K Ghosh, and B K Tripathy. On the theory of bags and lists. Inf. Sci.,

132:241–254, 2001.
[7] D Singh, A M Ibrahim, T Yohanna, and J N Singh. An overview of the applications

of multisets. Novi Sad J. Math., 37(2):73–92, 2007.
[8] A Syropoulos. Mathematics of multisets. Springer-Verlag, Berlin, Heidelberg, 347-

358, 2001.
[9] T O William-West, P A Ejegwa, and A U Amaonyeiro. On dynamic multisets and

their operations. Ann. Commun. Math., 4(3):284–292, 2021.
[10] L A Zadeh. Fuzzy sets. Inf. Contr., 8(3):338–353, 1965.
[11] A Rosenfeld. Fuzzy groups. J. Math. Anal. Appl., 35(3):512–517, 1971.
[12] J M Anthony and H Sherwood. A characterization of fuzzy subgroups. Fuzzy Set

Syst., 7:297–305, 1982.
[13] P A Ejegwa and J A Otuwe. Frattini fuzzy subgroups of fuzzy groups. J. Uni. Math.,

2(2):175–182, 2019.
[14] A Kumari, P Preeti, A Sehgal, P K Sharma, and S Sarita. Fuzzy subgroups of

non–abelian group zpn ⋉ zp for any prime p. AIP Conf. Proceedings, 2142:170025,
2019.

[15] A Sehgal and J Manjeet. The number of fuzzy subgroups for finite abelian p-group
of rank three. Adv. Fuzzy Math., 12(4):1035–1045, 2017.

[16] A Sehgal, S Sarita, and P K Sharma. The number of fuzzy subgroups of a finite
dihedral. Int. J. Fuzzy Math. Archive, 8(1):51–57, 2015.

[17] Sk Nazmul, P Majumdar, and S K Samanta. On multisets and multigroups. Ann.
Fuzzy Math. Inform., 6(3):643–656, 2013.

[18] P A Ejegwa. A study of multigroup structure and its acting principles on multiset.
PhD Thesis, Ahmadu Bello University, Zaria, Nigeria, 2018.

[19] J A Awolola and A M Ibrahim. Some results on multigroups. Quasi. Related Syst.,
24(2):169–177, 2016.

[20] P A Ejegwa and A M Ibrahim. Some homomorphic properties of multigroups. Bul.
Acad. Stiinte Repub. Mold. Mat., 83(1):67–76, 2017.

[21] U Adamu and M A Ibrahim. Strongly invariant submultigroups. Theory Appl. Math.
Computer Sci., 10(2):96–103, 2020.

[22] P A Ejegwa and A M Ibrahim. Normal submultigroups and comultisets of a multi-
group. Quasi. Related Syst., 25(2):231–244, 2017.

[23] A M Ibrahim and P A Ejegwa. Characteristic submultigroups of a multigroup. Gulf
J. Math., 5(4):1–8, 2017.

[24] J A Otuwe and M A Ibrahim. Frattini submultigroups of multigroups. Ratio Mathe-
matica, 39:147–163, 2020.

[25] J A Awolola. On multiset relations and factor multigroups. South East Asian J.
Math. Mathemat. Sci., 15(3):1–10, 2019.



P. A. Ejegwa et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5889 13 of 13

[26] J A Awolola. On cyclic multigroup family. Ratio Mathematica, 37:61–68, 2019.
[27] J A Awolola and P A Ejegwa. On some algebraic properties of order of an element

of a multigroup. Quasi. Related Syst., 25(1):21–26, 2017.
[28] P A Ejegwa and A M Ibrahim. On comultisets and factor multigroups. Theory Appl.

Math. Computer Sci., 7(2):124–140, 2017.
[29] P A Ejegwa and A M Ibrahim. Direct product of multigroups and its generalization.

Int. J. Math. Combin., 4:1–18, 2017.
[30] P A Ejegwa and A M Ibrahim. Some group’s analogous results in multigroup setting.

Ann. Fuzzy Math. Inform., 17(3):231–245, 2019.
[31] P A Ejegwa and A M Ibrahim. Some properties of multigroups. Palestine J. Math.,

9(1):31–47, 2020.
[32] A M Ibrahim and P A Ejegwa. Multigroup actions on multiset. Ann. Fuzzy Math.

Inform., 14(5):515–526, 2017.
[33] P A Ejegwa and J M Agbetayo. Some results on commutators in multigroup frame-

work. Ramanujan Soc. Math. Mathemat. Sci., 7(2):67–82, 2020.
[34] P A Ejegwa and A M Ibrahim. On divisible and pure multigroups and their properties.

Open J. Math. Sci., 4:377–385, 2020.
[35] S Debnath and A Debnath. Study of ring structure from multiset context. Applied

Sci., 21:84–90, 2019.
[36] P A Ejegwa. Concept of anti multigroups and its properties. Earthline J. Math. Sci.,

4(1):83–97, 2020.
[37] P Suma and S J John. Multiset approach to algebraic structures, In: Jana, C.,

Senapati, T. Pal, M. (Eds.); Handbook of Research on Emerging Applications of
Fuzzy Algebraic Structures. IGI Global Publisher, Hershey, Pennsylvania 17033-1240,
USA, 78-90, 2020.

[38] B C Tripathy, S Debnath, and D Rakshit. On multiset group. Proyecciones J. Math.,
37(3):479–489, 2018.

[39] P A Ejegwa, J M Agbetayo, J A Agba, and I M Adamu. Solvable multigroup and its
properties. Bull. Int. Math. Virtual Inst., 13(2):375–381, 2023.