KnE Engineering | International Conference on Basic Sciences and Its Applications (ICBSA-2018) | pages: 266–274

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1. Introduction

First we review that quaternion group, denoted by Q8 , was obtained based on the calculation of quaternions a+bi+cj+dk . Quaternions were first described by William Rowan Hamilton on October 1843 [1]. The quaternion group is a non-abelian group of order eight.

2. Materials and Methods

Here we provide Definitions of quaternion group dan matrix representation.

<statement> <title>Definition 1.1</title>

</statement>

[2] The quaternion group, Q8 , is defined by

Q8={1,1,i,i,j,j,k,k}

(1) with product computed as follows:

  • 1·a=a·1=a , for all aQ8

  • -1·a=a·-1=-a , for all aQ8

  • i·i=j·j=k·k=-1

  • i·j=k , j·i=-k

  • j·k=i , k·j=-i

  • k·i=j , i·k=-j

For every a,bQ8 , a·bb·a . Thus Q8 is a non-abelian group.

<statement> <title>Definition 1.2</title>

</statement>

[3] A matrix representation of degree n of a group G is a homomorphism ρ of G into general linear group GLn (F) over a field F.

It means that for every xG there corresponds an n×n matrix ρx with entries in F, and for all x,yG , [4]

ρxy=ρxρy.

(2)

Quaternion group Q8 can be represented by matrices, i.e. matrices of general linear group GL2 (C) over complex vector space C.

According to Marius Tarnauceanu [5], quaternion group is usually defined as a subgroup of the general linear group GL2 (C) consisting of 2×2 matrices with unit determinant called special linear group SL2 (C).

A homomorphism ρ:Q8SL2 (C) of quaternion group Q8 into the special linear group SL2 (C) over a complex vector space is given by:

1100111001
ii00iii00i
j0110j0110
k0ii0k0ii0

Since all of the matrices above have unit determinant, the homomorphism ρ is the representation of quaternion group into SL2 (C) under matrix mutiplication.

Suppose Q=I,-I,A,-A,B,-B,C,-C be a representation of quaternion group given by:

ρ1=1001=Iρ1=1001=I
ρi=i00i=Aρi=i00i=A
ρj=0110=Bρj=0110=B
ρk=0ii0=Cρk=0ii0=C.

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.

<statement> <title>Definition 1.3</title>

</statement>

[6] A non-abelian Dedekind group is called a Hamiltonian group.

Let H be a normal subgroup of G. For any aG , the set aH=ahhH is called the left coset of G in H. And the set Ha=hahH 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 G/H .

<statement> <title>Definition 1.4</title>

</statement>

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

G=N0N1N2Ni=e

(3) such that each Ni is normal in Ni-1 and the factor group Ni-1/Ni is abelian.

<statement> <title>Definition 1.5</title>

</statement>

[7] If subgroup HG and subgroup KG , then commutator subgroup

H,K=h,khH and kK

(4) where h,k is the commutator hkh-1k-1 .

Let the (ascending) central series of a finite group G be the sequence of subgroups

e=Z0(G)Z1(G)Z2(G).

(5) And the characteristic subgroups Zi(G) of group G is defined by induction:

Z1(G)=G;Zi+1(G)=Zi(G),G for i1.

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

<statement> <title>Definition 1.6</title>

</statement>

[7] A group G is called nilpotent if there is an integer c such that Zc+1(G)=e , and the least such c is called the class of the nilpotent group G.

<statement> <title>Definition 1.7</title>

</statement>

[8] A group G is called cyclic if G can be generated by an element xG such that G=xnnZ , n is an element of integers.

Such an element x is called a generator of G . G is a cylic group generated by x is indicated by writting G=x .

<statement> <title>Definition 1.8</title>

</statement>

[9] A group G is metacyclic if it contains a cyclic normal subgroup N such that G/N 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.

<statement> <title>Proposition 1</title>

</statement>

Representation of quaternion group is Hamiltonian.

Proof. Let Q=I,-I,A,-A,B,-B,C,-C be a representation of quaternion group. There are six normal subgroups of representation of quaternion group, which are N1=I , N2=I,-I , N3=I,-I,A,-A , N4=I,-I,B,-B , N5=I,-I,C,-C , and N6=I,-I,A,-,B,-B,C,-C .

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 Q is Hamiltonian.

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

<statement> <title>Proposition 2</title>

</statement>

Representation of quaternion group is solvable.

Proof. Let Q=I,-I,A,-A,B,-B,C,-C be a representation of quaternion group. One of the normal series for Q is N6N3N2N1 in which N2ΔN1 , N3ΔN2 , and N6ΔN3 . There are three factor groups in that normal series, which are N2/N1 , N3/N2 , and N6/N3 .

For all matrices X,YN2/N1 , XY=YX under matrix multiplication. Hence N1/N2 is abelian. For all matrices X,YN3/N2 , XY=YX under matrix multiplication. Hence N3/N2 is abelian. And for all matrices X,YN6/N3 , XY=YX under matrix multiplication. Hence N6/N3 is abelian. Hence all of the factor groups in normal series N6N3N2N1 of Q are abelian.

According to Definition 1.4, since there exists a normal series from Q such that each factor group is abelian, thus Q is solvable.

<statement> <title>Proposition 3</title>

</statement>

The representation of quaternion group is nilpotent.

Proof. Let Q=I,-I,A,-A,B,-B,C,-C be a representation of quaternion group. The sequence of subgroups of Q in given by I=Z0QZ1QZ2Q . And the characteristic subgroups Zi(Q) is defined by induction Z1(Q)=Q , and Zi+1(Q)=Zi(Q),Q .

Thus we have the following results:

  • We have Z2(Q)=Z1(Q),Q=Q,Q . Notice that the commutator subgroup Q,Q is the set of all commutator X,Y=XYX-1Y-1 where matrix XQ and matrix YQ . Hence we have Z2(Q)=N2 .

  • Next we have Z3(Q)=Z2(Q),Q=N2,Q .

Hence we have Z3(Q)=I .

According to Definition 1.6, since there is an integer c=2 such that Z3(Q)=Z2+1(Q)=I , thus Q is nilpotent, and the class of the nilpotent group Q is 2.

<statement> <title>Proposition 4</title>

</statement>

Representation of quaternion group is metacyclic.

Proof. Let Q=I,-I,A,-A,B,-B,C,-C be a representation of quaternion group. Based on Proposition 1, normal subgroups of representation of quaternion group Q are N1=I , N2=I,-I , N3=I,-I,A,-A , N4=I,-I,B,-B , N5=I,-I,C,-C , and N6=I,-I,A,-A,B,-B,C,-C .

Generators of normal subgroups of representation of quaternion group Q can be described as follows:

  • We have N1=I , hence N1 is cyclic.

  • We have N2=-I , hence N2 is cyclic.

  • We have N3=A and N3=-A , hence N3 is a cyclic group which has two generators.

  • We have N4=B and N4=-B , hence N4 is a cyclic group which has two generators.

  • We have N5=C and N5=-C , hence N5 is a cyclic group which has two generators.

In the other hand, normal subgroup N6 is not cyclic because N6X for any matrix XN6 .

Next, the factor group Q/N of normal subgroups of representation of quaternion group Q can be described as follows:

  • Factor group Factor group Q/N1 is not cyclic because Q/N1X for any matrix XQ/N1 .

  • Factor group Factor group Q/N2 is not cyclic because Q/N2X for any matrix XQ/N2 .

  • Factor group We have Q/N3=BN3 , hence factor group Q/N3 is cyclic.

  • Factor group We have Q/N4=IN4 , hence factor group Q/N4 is cyclic.

  • Factor group We have Q/N5=IN5 , hence factor group Q/N5 is cyclic.

  • Factor group We have Q/N6=IN6 , hence factor group Q/N6 is cyclic.

Thus, representation of quaternion group Q contains cyclic normal subgroups N3 , N4 , and N5 such that factor groups Q/N3 , Q/N4 , and Q/N5 are also cyclic. According to Definition 1.9, then Q is metacyclic.

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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