EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1717-1726
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On The Representations and Characters of Quaternions
Group Q8

Abdallah Labsir1,∗, El Mokhtar Fanich2

1 Laboratory of Applied Mathematics and Intelligent Systems Engineering (MAISI),
National School of Applied Sciences, Ibn Zohr University, Agadir, Morocco
2 Laboratory of Analysis, Modeling and Simulation, Department of Mathematics and
Computer Science, Ben M’sick’s Faculty of Sciences, Hassan II University of Casablanca,
Casablanca, Morocco

Abstract. We are interested in studying the linear representations of the Quaternions group Q8

by determining it’s charater table and irreducible representations which allows us to construct
some representations of degre 6 and 8 of Q8.

2020 Mathematics Subject Classifications: 20C15, 20C20, 20C30, 20G20

Key Words and Phrases: Irreducible representation, Character table, Quaternions group,
Group Q8

1. Introduction

Group representation theory allows the study of abstract groups by representing their
elements by invertible matrices. We then have the methods of linear algebra which often
make the study of these groups easier and make it possible to obtain new properties. The
idea is to make a group G act on a vector space V such that the action of each element is
compatible with the structure of the vector space, that is to say it is an element of GL(V)
the group of linear automorphisms of V and more only a bijection of V on V .

This concept emerges at the end of the 19th century and the general study of the
representations of a group is largely developed by William Burnside and Ferdinand Georg
Frobenius at the beginning of the 20th century[3].

Representation theory has enabled remarkable advances, first of all in group theory.
In particular, it plays a fundamental role in the classification theorem of finite simple
groups[5].

The theory of representations of general groups (not necessarily finite) and algebras also
has numerous applications in crystallographic chemistry, in engineering and especially in

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5175

Email addresses: abdallah.labsir@edu.uiz.ac.ma (A. Labsir),
elmokhtar.fanich-etu@etu.univh2c.ma (E. Fanich)

https://www.ejpam.com 1717 © 2024 EJPAM All rights reserved.



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1718

quantum physics: the theory of representation makes it possible to analyze the symmetries
of a related physical problem with a group by classifying the solutions of this problem
according to the irreducible representations[2].

In this work, we recall first some results on the theory of reprentations and character
of finite groups, and by the end we arrive to determine the irreducible representations and
to draw up the character table of the quaternions group Q8 which allows us to construct
some representations of degre 6 and 8 of this group.

2. Preleminaries

Let E be a vector space on K, where K = R or C, and let GL(E) the group of
isomorphisms of E on itself.

Definition 1 (See [3]). A representation of dimension n of a group G is the data of a
complex vector space E of dimension n, and of a morphism of groups

τ : G → GL(E)

For example, the standard representation of Sn is given as follows [4]:

τ : Sn → GLn(C) τσ(ei) = eσ(i)

We obtain the matrix of τσ by permuting the columns of the identity matrix in accordance
to σ. For example, for n = 3, we have:

τ(12) =

 0 1 0
1 0 0
0 0 1

 and τ(123) =

 0 0 1
1 0 0
0 1 0


Furthermore, if t ∈ S3 is the transposition 123 → 132 and c be the circular permutation

123 → 231 which generate S3. We set j = e
2iπ
3 . We can represent S3 in C2 by setting:

τ(e) = I, τ(t) =

[
0 1
1 0

]
, τ(c) =

[
j 0
0 j2

]
In particular, if the vector space E is of dimension n = |G| with basis indexed by the

elements of G, the representation is called the regular representation of G.

Definition 2 (See [3]). Let τ : G → GL(E) be a linear representation, and F a vector sub-
space of E stable for the operations of G, then τF : G → GL(F ) is a linear representation
of G in F ; called a sub-representation of E.

In addition, (τ, E) is called irreducible if E is not reduced to 0, and if no vector subspace
of E is stable by G, except 0 and E. And we have the following theorem:

Theorem 1 (See [2], Maschke theorem). every representation is a direct sum of irreducible
representations.



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1719

Recall here the notion of the character which is a function on G with complex values
characterizing the representation.

Definition 3 (See [1]). Let τ : G → GL(E) be a linear representation of a finite group G
in the vector space E. The character of G is defined as follow:

For all g ∈ G, χτ (g) = Tr(τg).

Furthermore, two representations of the same character are isomorphic, and we have:

Proposition 1 (See [2]). The character RG of the regular representation is given by:

RG(1) = |G|, and RG(g) = 0 if g ̸= 1.

The importance of the regular representation lies in the fact that an irreducible repre-
sentation Fi is contained in it a number of times equal to its degree ni.

Proposition 2 (See [2]). The degrees ni verify the relation
∑

n2
i = |G|.

Thus, we define the character table of a finite group G as follows:
Let c = |Conj(G)| the number of conjugation classes of G. The character map of G

is an array c × c of which the entries are the values of the irreducible characters on the
conjugation classes of G, the coefficient at the intersection of the column corresponding
to the character χ of the line corresponding to the conjugation class C, being χ(C). It’s
sort of the card of the group G[3].

For example, the group {±1} has two conjugation classes 1 and -1, and two irreducible
characters 1 and χ (of dimension 1 since {±1} is commutative); its character table is very
easy to establish:

1 -1

χ1 1 1

χ2 1 -1

The example of the group {±1} is a little too trivial to give an idea of how we can
construct the character table of a group. There are other groups when we use other
techniques (Burnside formula, character orthogonality relationships, etc.) to establish
character tables. For example the character table of the symmetric group σ3 is given as
follows :

First recall that:

σ3 = {1, (12), (23), (13), (123), (132)}

σ3 has 3 characters because that is its number of conjugation classes. First there is the
trivial character χ1. Then the morphism χ2 given by the signature of the elements of σ3.
Then by Burnside’s formula we obtain that the third character χ3 is such that n3 = 2.
We therefore obtain for the moment the following character table[1].



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1720

1 (12) (123)

χ1 1 1 1

χ2 1 -1 1

χ3 2 a b

To have a and b just use the orthogonality of the character table columns. So we get a = 0
and b = −1.

Hence finally:

1 (12) (123)

χ1 1 1 1

χ2 1 -1 1

χ3 2 0 −1

3. Representations and the characters of the quaternion group Q8

Recall that the non-commutative field H of quaternions can be obtained from the
field C by the construction of Cayley-Dickson: we provide the set C2 of pairs (z, w) of
complex numbers of the following addition and multiplication:

(z, w) + (z′, w′) = (z + z′, w + w′)

(z, w).(z′, w′) = (zz′ − w′w,w′z + wz′)

Note that this construction carried out from the field R of the real numbers produces
the field C.

The application z → (z, 0) allows us to canonically identify C to a subfield of H.
We set j = (0, 1) we have j2 = −1 and jz = zj for all z ∈ C (in particular k = ij =

−ji).
Any element h = (z, w) of H is then uniquely written h = z+ jw with z, w ∈ C so that

(1, j) is a basis of the C-vector space H while (1, i, j, k) is a basis of the R-vector space H.
We have: 

i2 = j2 = k2 = −1
ij = −ji = k
jk = −kj = i
ki = −ik = j

We again set h = z − jw so that h = h and hh′ = h′h for all h, h′ ∈ H. We then have
N(h) = hh̄ ∈ R+ and N(hh′) = N(h)N(h′). Then all h ∈ H∗ is invertible and we have

h−1 =
1

N(h)
h.



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1721

Finally, we have the canonical representation of the R-algebra H:

π : H → M2(C)

h = z + jw →
(

z −w
w z

)
Note that we have N(h) = det(π(h)) so that the matrix π(h) is invertible if and only if
h ̸= 0. In particular, we have:

π(±1) = ±
(

1 0
0 1

)
π(±i) = ±

(
i 0
0 −i

)

π(±j) = ±
(

0 −1
1 0

)
π(±k) = ±

(
0 −i
−i 0

)
The quaternion group :

Q8 =< i, j >= {1,−1, i,−i, j,−j, k,−k}

The group Q8 has 5 conjugation classes:

K1 = {1} K2 = {−1} K3 = {i,−i} K4 = {j,−j} K5 = {k,−k}

3.1. Character table of group Q8

We have the group {±1} is distinguished in Q8 and Q8/{±1} ≈ V (V : Klein group)
and according to the Burnside formula (n2

1 + n2
2 + n2

3 + n2
4 + n2

5 = 8), we will have n5 = 2
So the character table of Q8 is of the form:

K1 K2 K3 K4 K5

χ1 1 1 1 1 1

χ2 1 1 1 -1 -1

χ3 1 1 -1 1 -1

χ4 1 1 -1 -1 1

χ5 2 a b c d

Then by orthogonality of the columns we obtain a = −2 et b = c = d = 0. Finally the
character table of Q8 is as follows:

K1 K2 K3 K4 K5

χ1 1 1 1 1 1

χ2 1 1 1 -1 -1

χ3 1 1 -1 1 -1

χ4 1 1 -1 -1 1

χ5 2 -2 0 0 0



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1722

3.2. Irreducible representations of the group Q8

3.2.1. Realization of irreducible representations of degree 1

Let τ : H → C∗ be a 1-dimensional representation of H. Then τ cannot be faithful,
otherwise H would be abelian.

Therefore, Kerτ is a distinguished subgroup of H, Kerτ ̸= {1}. If Kerτ = H we
obtain the trivial representation τ1. Others possibilities for Kerτ are:

Z(H) = {±1}, I = {±1,±i}

J = {±1,±j}, K = {±1,±k}

But, H/I ≈ H/J ≈ H/K ≈ Z2 and H/Z(H) ≈ Z2 × Z2. Obviously, τ is factorized as the
composite of the projection on the quotient H/Kerτ → C∗. Furthermore, there is no only
one way to represent Z2 in a non-trivial way in C and therefore we obtain the following
representations:

When kerτ = I

τ2({±1,±i}) = 1, τ2({±j,±k}) = −1

When kerτ = J

τ3({±1,±j}) = 1, τ3({±i,±k}) = −1

When kerτ = K

τ4({±1,±k}) = 1, τ4({±i,±j}) = −1

When kerτ = Z(H), the representations obtained by composing the projection on the
quotient by a representation of E in C∗ still give the three representations above. The
four representations thus found are not isomorphic with each other.

3.2.2. Realization of the irreducible representation of degree 2

Q8 operates by left multiplication on the space H of quaternions, which is a vector space
on the right on C = R ⊕ Ri, of dimension 2. In the basis {1, j}, we obtain the following
matrix representation:

τ5(±1) = ±
(

1 0
0 1

)
τ5(±i) = ±

(
i 0
0 −i

)

τ5(±j) = ±
(

0 −1
1 0

)
τ5(±k) = ±

(
0 −i
−i 0

)



A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1723

3.3. Some representations of the group Q8

We know that any representation is the sum of irreducible representations, therefore,
from these we can construct the desired representations.

3.3.1. The regular representation of Q8

We have |Q8| = 8, therefore the degree of the regular representation equal to 8, and we
know that every irreducible representation is contained in the regular representation a
number of times equal to its degree, therefore:

τreg = τ1 ⊕ τ2 ⊕ τ3 ⊕ τ4 ⊕ 2τ5

so:

τreg : Q8 → GL8(C)

defined by :

τreg(1) =



1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1



τreg(−1) =



1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 −1 0 0 0
0 0 0 0 0 −1 0 0
0 0 0 0 0 0 −1 0
0 0 0 0 0 0 0 −1



τreg(i) =



1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 −1 0 0 0 0 0
0 0 0 −1 0 0 0 0
0 0 0 0 i 0 0 0
0 0 0 0 0 −i 0 0
0 0 0 0 0 0 i 0
0 0 0 0 0 0 0 −i





A. Labsir, E. Fanich / Eur. J. Pure Appl. Math, 17 (3) (2024), 1717-1726 1724

τreg(−i) =



1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 −1 0 0 0 0 0
0 0 0 −1 0 0 0 0
0 0 0 0 −i 0 0 0
0 0 0 0 0 i 0 0
0 0 0 0 0 0 −i 0
0 0 0 0 0 0 0 i



τreg(j) =



1 0 0 0 0 0 0 0
0 −1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 −1 0 0 0 0
0 0 0 0 0 −1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 −1
0 0 0 0 0 0 1 0



τreg(−j) =



1 0 0 0 0 0 0 0
0 −1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 −1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 −1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 −1 0



τreg(k) =



1 0 0 0 0 0 0 0
0 −1 0 0 0 0 0 0
0 0 −1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 −i 0 0
0 0 0 0 −i 0 0 0
0 0 0 0 0 0 0 −i
0 0 0 0 0 0 −i 0



τreg(−k) =



1 0 0 0 0 0 0 0
0 −1 0 0 0 0 0 0
0 0 −1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 i 0 0
0 0 0 0 i 0 0 0
0 0 0 0 0 0 0 i
0 0 0 0 0 0 i 0





REFERENCES 1725

3.3.2. The representation of degree 6 of Q8

Let θ be the representation of degree 6 of Q8, then:

θ = τ1 ⊕ τ2 ⊕ τ3 ⊕ τ4 ⊕ τ5

so:

θ : Q8 → GL6(C)

defined by :

θ(1) =



1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1

 ; θ(−1) =



1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 −1 0
0 0 0 0 0 −1



θ(i) =



1 0 0 0 0 0
0 1 0 0 0 0
0 0 −1 0 0 0
0 0 0 −1 0 0
0 0 0 0 i 0
0 0 0 0 0 −i

 ; θ(−i) =



1 0 0 0 0 0
0 1 0 0 0 0
0 0 −1 0 0 0
0 0 0 −1 0 0
0 0 0 0 −i 0
0 0 0 0 0 i



θ(j) =



1 0 0 0 0 0
0 −1 0 0 0 0
0 0 1 0 0 0
0 0 0 −1 0 0
0 0 0 0 0 −1
0 0 0 0 1 0

 ; θ(−j) =



1 0 0 0 0 0
0 −1 0 0 0 0
0 0 1 0 0 0
0 0 0 −1 0 0
0 0 0 0 0 1
0 0 0 0 −1 0



θ(k) =



1 0 0 0 0 0
0 −1 0 0 0 0
0 0 −1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 −i
0 0 0 0 −i 0

 ; θ(−k) =



1 0 0 0 0 0
0 −1 0 0 0 0
0 0 −1 0 0 0
0 0 0 1 0 0
0 0 0 0 0 i
0 0 0 0 i 0


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[2] David Renard. Groups and representation. Ecole Polytechnique, 2010.



REFERENCES 1726

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