Some Properties of Representation of Quaternion Group

ICBSA 2018 International Conference on Basic Sciences and Its Applications Volume 2019

Conference Paper Some Properties of Representation of Quaternion Group Sri Rahayu, Maria Soviana, Yanita, and Admi Nazra Department of Mathematics, Andalas University, Limau Manis, Padang, 25163, Indonesia

Abstract The quaternions are a number system in the form 푎 + 푏푖 + 푐푗 + 푑푘. The quaternions ±1, ±푖, ±푗, ±푘 form a non-abelian group of order eight called quaternion group. Quaternion

group can be represented as a subgroup of the general linear group 퐺퐿2(C). In this paper, we discuss some group properties of representation of quaternion group related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group.

Keywords: representation of quaternion group, hamiltonian group, solvable group, nilpotent group, metacyclic group

Corresponding Author: Sri Rahayu [email protected]

Received: 19 February 2019 1. Introduction Accepted: 5 March 2019 Published: 16 April 2019 First we review that quaternion group, denoted by 푄8, was obtained based on the Publishing services provided by calculation of quaternions 푎 + 푏푖 + 푐푗 + 푑푘. Quaternions were ﬁrst described by William Knowledge E Rowan Hamilton on October 1843 [1]. The quaternion group is a non-abelian group of Sri Rahayu et al. This article is order eight. distributed under the terms of the Creative Commons Attribution License, which 2. Materials and Methods permits unrestricted use and redistribution provided that the original author and source are Here we provide Deﬁnitions of quaternion group dan matrix representation. credited.

Selection and Peer-review under Deﬁnition 1.1. [2] The quaternion group, 푄8, is deﬁned by the responsibility of the ICBSA Conference Committee. 푄8 = {1, − 1, 푖, − 푖, 푗, − 푗, 푘, − 푘} (1)

with product computed as follows:

• 1 ⋅ 푎 = 푎 ⋅ 1 = 푎, for all 푎 ∈ 푄8

• (−1) ⋅ 푎 = 푎 ⋅ (−1) = −푎, for all 푎 ∈ 푄8 • 푖 ⋅ 푖 = 푗 ⋅ 푗 = 푘 ⋅ 푘 = −1

How to cite this article: Sri Rahayu, Maria Soviana, Yanita, and Admi Nazra, (2019), “Some Properties of Representation of Quaternion Group” in Page 266 International Conference on Basic Sciences and Its Applications, KnE Engineering, pages 266–274. DOI 10.18502/keg.v1i2.4451 ICBSA 2018

• 푖 ⋅ 푗 = 푘, 푗 ⋅ 푖 = −푘

• 푗 ⋅ 푘 = 푖, 푘 ⋅ 푗 = −푖

• 푘 ⋅ 푖 = 푗, 푖 ⋅ 푘 = −푗

For every 푎, 푏 ∈ 푄8, 푎 ⋅ 푏 ≠ 푏 ⋅ 푎. Thus 푄8 is a non-abelian group.

Deﬁnition 1.2. [3]A matrix representation of degree 푛 of a group G is a homomorphism

휌 of G into general linear group 퐺퐿푛(F) over a ﬁeld F. It means that for every 푥 ∈ 퐺 there corresponds an 푛 × 푛 matrix 휌 (푥) with entries in F, and for all 푥, 푦 ∈ 퐺,[4]

휌 (푥푦) = 휌 (푥) 휌 (푦) . (2)

Quaternion group 푄8 can be represented by matrices, i.e. matrices of general linear

group 퐺퐿2(C) over complex vector space C. According to Marius Tarnauceanu [5], quaternion group is usually deﬁned as a sub-

group of the general linear group 퐺퐿2(C) consisting of 2 × 2 matrices with unit determi-

nant called special linear group 푆퐿2(C).

A homomorphism 휌 ∶ 푄8 → 푆퐿2(C) of quaternion group 푄8 into the special linear

group 푆퐿2(C) over a complex vector space is given by:

⎛ ⎞ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ −1 0 ⎟ 1 ↦ ⎜ ⎟ − 1 ↦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 1 ⎠ ⎝ 0 −1 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 푖 0 ⎟ ⎜ −푖 0 ⎟ 푖 ↦ ⎜ ⎟ − 푖 ↦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 −푖 ⎠ ⎝ 0 푖 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 0 1 ⎟ ⎜ 0 −1 ⎟ 푗 ↦ ⎜ ⎟ − 푗 ↦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ −1 0 ⎠ ⎝ 1 0 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 0 푖 ⎟ ⎜ 0 −푖 ⎟ 푘 ↦ ⎜ ⎟ − 푘 ↦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 푖 0 ⎠ ⎝ −푖 0 ⎠ Since all of the matrices above have unit determinant, the homomorphism 휌 is the

representation of quaternion group into 푆퐿2(C) under matrix mutiplication.

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Suppose 푄 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶} be a representation of quaternion group given by: ⎛ ⎞ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ −1 0 ⎟ 휌 (1) = ⎜ ⎟ = 퐼 휌 (−1) = ⎜ ⎟ = −퐼 ⎜ ⎟ ⎜ ⎟ ⎝ 0 1 ⎠ ⎝ 0 −1 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 푖 0 ⎟ ⎜ −푖 0 ⎟ 휌 (푖) = ⎜ ⎟ = 퐴 휌 (−푖) = ⎜ ⎟ = −퐴 ⎜ ⎟ ⎜ ⎟ ⎝ 0 −푖 ⎠ ⎝ 0 푖 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 0 1 ⎟ ⎜ 0 −1 ⎟ 휌 (푗) = ⎜ ⎟ = 퐵 휌 (−푗) = ⎜ ⎟ = −퐵 ⎜ ⎟ ⎜ ⎟ ⎝ −1 0 ⎠ ⎝ 1 0 ⎠

⎛ ⎞ ⎛ ⎞ ⎜ 0 푖 ⎟ ⎜ 0 −푖 ⎟ 휌 (푘) = ⎜ ⎟ = 퐶 휌 (−푘) = ⎜ ⎟ = −퐶. ⎜ ⎟ ⎜ ⎟ ⎝ 푖 0 ⎠ ⎝ −푖 0 ⎠ Here we present some Deﬁnitions related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group. A Dedekind group is a group in which every subgroup is normal. Every subgroup in an abelian group is normal, hence all abelian groups are Dedekind. But there also exists non-abelian group in which all of its subgroup are normal.

Deﬁnition 1.3. [6]A non-abelian Dedekind group is called a Hamiltonian group. Let H be a normal subgroup of G. For any 푎 ∈ 퐺, the set 푎퐻 = {푎ℎ | ℎ ∈ 퐻 }is called the left coset of G in H. And the set 퐻푎 = {ℎ푎 | ℎ ∈ 퐻 }is called the right coset of G in H. Let H be a normal subgroup of G, then the set of right (or left) cosets of H in G is itself a group called the factor group of G by H , denoted by 퐺/퐻.

Deﬁnition 1.4. [3] A group G is called solvable if there exist a normal series from group G

퐺 = 푁0 ⊃ 푁1 ⊃ 푁2 ⊃ ⋯ ⊃ 푁푖 = {푒} (3)

such that each 푁푖 is normal in 푁푖−1 and the factor group 푁푖−1/푁푖 is abelian.

Deﬁnition 1.5. [7] If subgroup퐻 ≤ 퐺 and subgroup퐾 ≤ 퐺, then commutator subgroup

[퐻, 퐾] = {[ℎ, 푘] |ℎ ∈ 퐻 and 푘 ∈ 퐾} (4)

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where [ℎ, 푘] is the commutator ℎ푘ℎ−1푘−1. Let the (ascending) central series of a ﬁnite group G be the sequence of subgroups

{푒} = 푍0(퐺) ⊂ 푍1(퐺) ⊂ 푍2(퐺) ⊂ ⋯ . (5)

And the characteristic subgroups 푍푖(퐺) of group 퐺is deﬁned by induction:

푍1(퐺) = 퐺; 푍푖+1(퐺) = [푍푖(퐺), 퐺] for 푖 ≥ 1. (6)

The commutator subgroup, characteristic subgroups 푍푖(퐺), and the central series of a group G lead to the following deﬁnition.

Deﬁnition 1.6. [7] A group G is called nilpotent if there is an integer 푐 such that 푍푐+1(퐺) = {푒}, and the least such 푐 is called the class of the nilpotent group G.

Deﬁnition 1.7. [8] A group G is called cyclic if 퐺can be generated by an element 푥 ∈ 퐺 such that 퐺 = {푥푛 | 푛 ∈ Z}, n is an element of integers. Such an element 푥 is called a generator of 퐺. 퐺 is a cylic group generated by 푥 is indicated by writting 퐺 = ⟨푥⟩.

Deﬁnition 1.8. [9] A group G is metacyclic if it contains a cyclic normal subgroup 푁 such that 퐺/푁 is also cyclic.

3. Result and Discussion

Here we present the results from our studies related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group. Some properties of representation of quaternion group are contained in some following Propositions.

Proposition 1. Representation of quaternion group is Hamiltonian.

Proof. Let 푄 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶} be a representation of quaternion group.

There are six normal subgroups of representation of quaternion group, which are 푁1 =

{퐼}, 푁2 = {퐼, −퐼}, 푁3 = {퐼, −퐼, 퐴, −퐴}, 푁4 = {퐼, −퐼, 퐵, −퐵}, 푁5 = {퐼, −퐼, 퐶, −퐶}, and

푁6 = {퐼, −퐼, 퐴, −, 퐵, −퐵, 퐶, −퐶}.

A Dedekind group is a group G such that every subgroup of G is normal. According to Deﬁnition 1.3, then representation of quaternion grup 푄 is Hamiltonian.

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Every subgroup is normal in every abelian group. In the other hand, 푄 is a non-abelian group in which every subgroup is normal.

Proposition 2. Representation of quaternion group is solvable.

Proof. Let 푄 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶} be a representation of quaternion group.

One of the normal series for 푄 is 푁6 ⊃ 푁3 ⊃ 푁2 ⊃ 푁1 in which 푁2Δ푁1, 푁3Δ푁2, and

푁6Δ푁3. There are three factor groups in that normal series, which are 푁2/푁1, 푁3/푁2,

and 푁6/푁3.

For all matrices 푋, 푌 ∈ 푁2/푁1, 푋푌 = 푌푋 under matrix multiplication. Hence 푁1/푁2

is abelian. For all matrices 푋, 푌 ∈ 푁3/푁2, 푋푌 = 푌푋 under matrix multiplication. Hence

푁3/푁2 is abelian. And for all matrices 푋, 푌 ∈ 푁6/푁3, 푋푌 = 푌푋 under matrix multi-

plication. Hence 푁6/푁3 is abelian. Hence all of the factor groups in normal series

푁6 ⊃ 푁3 ⊃ 푁2 ⊃ 푁1 of 푄 are abelian. According to Deﬁnition 1.4, since there exists a normal series from 푄 such that each factor group is abelian, thus 푄 is solvable.

Proposition 3. The representation of quaternion group is nilpotent.

Proof. Let 푄 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶} be a representation of quaternion group.

The sequence of subgroups of 푄 in given by {퐼} = 푍0 (푄) ⊂ 푍1 (푄) ⊂ 푍2 (푄) ⊂ ⋯. And

the characteristic subgroups 푍푖(푄)is deﬁned by induction 푍1(푄) = 푄, and 푍푖+1(푄) =

[푍푖(푄), 푄]. Thus we have the following results:

• We have 푍2(푄) = [푍1(푄), 푄] = [푄, 푄]. Notice that the commutator subgroup [푄, 푄] is the set of all commutator [푋, 푌 ] = 푋푌푋−1푌 −1 where matrix 푋 ∈ 푄 and matrix 푌 ∈ 푄.

[푄, 푄] = { [푋, 푌 ] | matrix X ∈ 푄, matrix Y ∈ 푄}

= {퐼, −퐼}

= 푁2

Hence we have 푍2(푄) = 푁2.

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• Next we have 푍3(푄) = [푍2(푄), 푄] = [푁2, 푄].

[푁2, 푄] = { [푋, 푌 ] | matrix X ∈ 푁2, matrix Y ∈ 푄}

= {퐼}

= 푁1

Hence we have 푍3(푄) = {퐼}.

According to Deﬁnition 1.6, since there is an integer 푐 = 2 such that 푍3(푄) = 푍2+1(푄) = {퐼}, thus 푄 is nilpotent, and the class of the nilpotent group 푄 is 2.

Proposition 4. Representation of quaternion group is metacyclic.

Proof. Let 푄 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶} be a representation of quaternion group. Based on Proposition 1, normal subgroups of representation of quaternion group 푄 are

푁1 = {퐼}, 푁2 = {퐼, −퐼}, 푁3 = {퐼, −퐼, 퐴, −퐴}, 푁4 = {퐼, −퐼, 퐵, −퐵}, 푁5 = {퐼, −퐼, 퐶, −퐶},

and 푁6 = {퐼, −퐼, 퐴, −퐴, 퐵, −퐵, 퐶, −퐶}. Generators of normal subgroups of representation of quaternion group 푄 can be described as follows:

•

푛 0 1 2 (퐼) = {… , (퐼) , (퐼) , (퐼) ,…}

= {퐼}

= 푁1

We have 푁1 = ⟨퐼⟩, hence 푁1 is cyclic. •

푛 0 1 2 3 (−퐼) = {… , (−퐼) , (−퐼) , (−퐼) , (−퐼) ,…}

= {퐼, −퐼}

= 푁2

We have 푁2 = ⟨−퐼⟩, hence 푁2 is cyclic. •

푛 0 1 2 3 (퐴) = {…, (퐴) , (퐴) , (퐴) , (퐴) ,…}

= {퐼, −퐼, 퐴, −퐴}

= 푁3

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푛 0 1 2 3 (−퐴) = {… , (−퐴) , (−퐴) , (−퐴) , (−퐴) ,…}

= {퐼, −퐼, 퐴, −퐴}

= 푁3

We have 푁3 = ⟨퐴⟩ and 푁3 = ⟨−퐴⟩, hence 푁3 is a cyclic group which has two generators.

•

푛 0 1 2 3 (퐵) = {… , (퐵) , (퐵) , (퐵) , (퐵) ,…}

= {퐼, 퐵, −퐼, −퐵}

= 푁4

푛 0 1 2 3 (−퐵) = {… , (−퐵) , (−퐵) , (−퐵) , (−퐵) ,…}

= {퐼, −퐵, −퐼, 퐵}

= 푁4

We have 푁4 = ⟨퐵⟩and 푁4 = ⟨−퐵⟩, hence 푁4 is a cyclic group which has two generators.

•

푛 0 1 2 3 (퐶) = {… , (퐶) , (퐶) , (퐶) , (퐶) ,…}

= {퐼, 퐶, −퐼, −퐶}

= 푁5

푛 0 1 2 3 (−퐶) = {… , (−퐶) , (−퐶) , (−퐶) , (−퐶) ,…}

= {퐼, −퐶, −퐼, 퐶}

= 푁5

We have 푁5 = ⟨퐶⟩ and 푁5 = ⟨−퐶⟩, hence 푁5 is a cyclic group which has two generators.

In the other hand, normal subgroup 푁6 is not cyclic because 푁6 ≠ ⟨푋⟩ for any matrix

푋 ∈ 푁6. Next, the factor group 푄/푁 of normal subgroups of representation of quaternion group 푄 can be described as follows:

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• Factor group

푄/푁1 = {푋푁1 | matrix 푋 ∈ 푄}

= {퐼푁1, −퐼푁1, 퐴푁1, −퐴푁1, 퐵푁1, −퐵푁1}

Factor group 푄/푁1 is not cyclic because 푄/푁1 ≠ ⟨푋⟩ for any matrix 푋 ∈ 푄/푁1. • Factor group

푄/푁2 = {푋푁2 | matrix 푋 ∈ 푄}

= {퐼푁2, 퐴푁2, 퐵푁2, 퐶푁2}

Factor group 푄/푁2 is not cyclic because 푄/푁2 ≠ ⟨푋⟩ for any matrix 푋 ∈ 푄/푁2. • Factor group

푄/푁3 = {푋푁3 | matrix 푋 ∈ 푄}

= {퐼푁3, 퐵푁3}

We have 푄/푁3 = ⟨퐵푁3⟩, hence factor group 푄/푁3 is cyclic. • Factor group

푄/푁4 = {푋푁4 | matrix 푋 ∈ 푄}

= {퐼푁4, 퐴푁4}

We have 푄/푁4 = ⟨퐼푁4⟩, hence factor group 푄/푁4 is cyclic. • Factor group

푄/푁5 = {푋푁5 | matrix 푋 ∈ 푄}

= {퐼푁5, 퐴푁5}

We have 푄/푁5 = ⟨퐼푁5⟩, hence factor group 푄/푁5 is cyclic. • Factor group

푄/푁6 = {푋푁6 | matrix 푋 ∈ 푄}

= {퐼푁6}

We have 푄/푁6 = ⟨퐼푁6⟩, hence factor group 푄/푁6 is cyclic.

Thus, representation of quaternion group 푄 contains cyclic normal subgroups 푁3, 푁4,

and 푁5 such that factor groups 푄/푁3, 푄/푁4, and 푄/푁5 are also cyclic. According to Deﬁnition 1.9, then 푄 is metacyclic.

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4. Conclusions

From our results, some properties of representation of quaternion group are proved to be contained in the Propositions above, which are:

• Representation of quaternion group is Hamiltonian.

• Representation of quaternion group is solvable.

• Representation of quaternion group is nilpotent.

• Representation of quaternion group is metacyclic

References

[1] Waerden, B. L. (1976). Hamilton’s Discovery of Quaternions. Mathematical Association of America. 49(5), 227-234. [2] Dummit, D. S. and Foote R. M. (2004). Abstract Algebra. Third Edition, John Wiley & Sons, New York. [3] Hall, M. (2004). The Theory of Groups. The Macmillan Company, New York. [4] Burrow, M. (1993). Representation Theory of Finite Group. Academic Press Inc., New York. [5] Tarnauceanu, M. (2013). A Characterization of The Quaternion Group. Mathematics Subject Classiﬁcation. 21(1), 209-214. [6] Nymann, D. S. (1967). Dedekind Groups. Paciﬁc Journal of Mathematics, 21(1), 153- 160. [7] Rotman, J. J. (1999). An Introduction to The Theory of Groups. Fourth Edition, Springer-Verlag, New York. [8] Herstein, I. N. (1975). Topics in Algebra. Second Edition, John Wiley & Sons, New York. [9] Hempel, C. E. (2000). Metacyclic Groups. Communications in Algebra, 28(8), 3865- 3897.

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