The Rational valued characters of the Group (Q2m×C2)

The Rational valued characters of the Group (Q2m×C2)

when m=2h , h∈Z+

                           

1.Abstract

       The main purpose of this paper is to find the rational valued characters table of the group (Q2m×C2) ,when m=2h , h∈Z+, which is denoted by ≡*(Q2h+1×C2) ,where Q2m is denoted to Quaternion group ,and C2  is the cyclic group of order 2.

Moreover we have found the general form of the rational valued characters table of the group (Q2h+1×C2).

2.Introduction

        Let G be a finite group ,two elements of G are said to be -conjugate  if the cyclic subgroups they generate are conjugate in G; this -relation defines an equivalence  relation on G.Its classes are called - classes

   The Z-valued class function on the group G, which is constant on the - classes forms a finitely generated abelian group        cf(G,Z) of a rank equal to the number of - classes

         The intersection of cf(G,Z) with the group of all generalized characters of G, R(G) is a normal subgroup of cf(G,Z) denoted by   , Each element in  can be written as u1θ1+ u2θ2+……+ u θ , where    is the number of

 -classes ,

 u1,u2 , … ,u  Z  and  = , where   is an irreducible character of the group G and   is any element in Galios group  .

    Let ≡*(G) denotes the   matrix which the

rows corresponds to the  's  and columns correspond to the - classes of G .In 1995

N. R. Mahamood [3] studied the factor group cf(Q2m,Z) / (Q2m). The aim of this paper  is to find ≡*(Q2h+1×C2) and determine general

( ) matrix form of the rational valued characters table of the group(Q2h+1×C2).

3.Preliminaries

The Generalized Quaternion Group Q2m(3.1)  [3]

          

    For each positive integer m, The generalized Quaternion Group Q2m of order 4m  with two  generators   and  satisfies

Q2m = 

which has the following properties

The character table of the quaternion group Q2m when m=2h , h∈Z+ (3.2) [3]

   There are two types of irreducible characters .one of them is the character of the linear representations R1, R2, R3 and R4 which are denoted by ψ1, ψ2, ψ3 and ψ4.

respectively  as in the following table:

                                                                    xk           xky

ψ1                                                                 1           1

ψ2                                                                 1           -1

ψ3                                                           (-1)k           (-1)k

Ψ4                                                           (-1)k           (-1)k+1

Table (1)

   Where  0 ≤ k ≤ 2m-1.

   The rest characters of irreducible representations Th of degree 2 are denoted by χh such that :

χh (xk) = ωhk+ω-hk

             =eπihk/m + e-πihk/m

                   =2cos(πhk/m)

    We are denoted to ωhk+ω-hk by Vhk ,Thus  Vhk =V2m-hk  ,Vm=-2  , V2M =2  .also we will write VJ(hk)  such that

J(hk) =min{hk (mod 2m), 2m-hk(mod 2m) },

in the character table of the quaternion group Q2m when m=2h, where VJ (hk)= 2cos(πJ(hk)/m),

χh (xky) = 0                                                     

       Where 0 ≤ k ≤ 2m-1, 1 ≤ h ≤ m-1

 and ω =e2πi/2m.

  So there are m+3 irreducible characters of Q2m. Then , the general form of the characters table of Q2m when m=2h , h∈Z+ is given in the table (2) .

Theorem (3.3)[1]

Let T1:G1→GL(n,K) and T2:G2→GL(m,K) are two irreducible representation of the group G1 and G2 with characters χ1 and χ2 respectively, then T1 T2 is irreducible representation of the group G1×G2 with the character χ1.χ2.

The Group Q2m×C2 (3.4)

The direct product group Q2m×C2 ,where C2 is a cyclic group of order 2 then         

                                              |Q2m×C2|= 8m .

Since , the irreducible representations of the group Q2m×C2  are the tensor products of those of Q2m and those of C2.The group C2 has two irreducible representations, their characters σ1 and σ2 are given in the table(2):

CLα                                                                1           r

| CLα|                                                            1           1

σ1                                                                  1           1

        σ2                                                          1           -1

    According to Theorem (3.3), each irreducible character χi of Q2m defines two irreducible characters χi1,χi2 such that χi1=χiσ1 , χi2 = χiσ2 of Q2m×C2  .

Then   ≡(Q2m×C2) = ≡(Q2m)   ≡(C2)

Example (3.5)

   To find characters table of Q16×C2 .

From (3.3) we have the character table of Q16 as Table (3),Where Vi  =2cos(πi/8) , V2m=2 ,

 Vm=-2 ,V4=2cos(4π/8)=0 .

And

CLα                                                                1           R

| CLα|                                                            1           1

σ1                                                                  1           1

σ2                                                                  1           -1

          

           

   By Theorem (3.3), the characters table of Q16×C2 can be written as follows:   

 ≡ (Q16×C2) = ≡ (Q16) ≡ (C2),

 Then ≡(Q16×C2) is given in the table (4) .

Where Vi  =2cos(πi/8) , V2m=2 , Vm=-2 ,V4=2cos(4π/8)=0 .

4.The main results

Proposition(4.1)[2]

     The rational valued characters   form basis for  , where   are the irreducible characters of G and their numbers are equal to the number of all distinct Γ-classes of G.

The rational character table of the quaternion group Q2m when m=2h , h∈Z+ (4.2)[3]

   the rational characters table of Q2m when m=2h , h∈Z+ is given in the following table (after  change order the rows and the columns) :

≡*(Q2h+1)=

1     1    1     1    1    .    .    .    1       1       1

-1     1   -1    1    1    .    .    .    1       1       1

-1    -1    1    1    1    .    .    .    1       1       1

 1    -1   -1    1    1    .    .    .    1       1       1

 0     0    0   -2    2    .    .    .    2       2       2

 0     0    0    0   -4    .    .    .    4       4       4

                                    

                                    

0      0     0   .    .    .   .   0     -2h-1    2h-1     2h-1

0      0     0   .    .    .   .   0       0     -2h       2h

Example (4.3)

 To construct the rational valued characters table of Q16×C2,when have to do the following:

From Example(3.5) we have the characters table of Q16×C2 .

By the definition of Q2m×C2 :

       ≡(Q16×C2)= (≡Q16)  (≡C2)

  

   To calculate the rational valued character table of Q16×C2,     

θ11=11, θ12=12, θ21=41, θ22=42, θ31=21, θ32=22, θ41=31, θ42=32  ,

 θ51= χ41, θ52= χ42  .

The elements of Gal(χ1i)/Q, are :

     {σ1i, σ3i, σ5i, σ7i }

  Where σ1i(χ1i)= χ1i , σ3i(χ1i)= χ3i ,

σ5i(χ1i)= χ5i , σ7i(χ1i)= χ7i  and i=1,2.

By proposition (4.1)

1- (I) if i=1

θ71= σ11(χ11)+ σ31(χ11)+ σ51(χ11)+ σ71(χ11)

θ71 ([1,1])=2+2+2+2=8

θ71 ([1,r])=2+2+2+2=8

θ71 ([x8,1])= (-2)+ (-2)+ (-2)+ (-2)=-8

 θ71 ([x8,r])= (-2)+ (-2)+ (-2)+ (-2)=-8

θ71([x,1])=V1+ V3+ V5+ V7=0

 θ71 ([x,r])= V1+ V3+ V5+ V7=0

θ71 ([x2,1])= V2+ V6+ V6+ V2=0

 θ71 ([x2,r])= V2+ V6+ V6+ V2=0

θ71 ([x4,1])= 0+0+0+0=0

θ71 ([x4,r])= 0+0+0+0=0

θ71([y,1])=0

θ71 ([y, r])=0

θ71 ([xy,1])=0

θ71 ([xy, r])=0

 1- (II) if i=2             

θ72= σ12(χ12)+ σ32(χ12)+ σ52(χ12)+ σ72(χ12)

 θ72 ([1,1])=2+2+2+2=8

θ72 ([1,r])= (-2)+ (-2)+ (-2)+ (-2)=-8

θ72 ([x8,1])= (-2)+ (-2)+ (-2)+ (-2)=-8

 θ72 ([x8,r])= 2+2+2+2=8

θ72 ([x,1])=V1+ V3+ V5+ V7=0

 θ72 ([x,r])= (-V1)+ (-V3)+ (-V5)+(-V7)=0

θ72 ([x2,1])= V2+ V6+ V6+ V2=0

 θ72 ([x2,r])= (-V2)+ (-V6)+ (-V6)+ (-V2)=0

θ72 ([x4,1])= 0+0+0+0=0

θ72 ([x4,r])= 0+0+0+0=0

θ72 ([y,1])=0

θ72([y, r])=0

θ72 ([xy,1])=0

θ72([xy, r])=0

Also σ1i(χ2i)= χ2i , σ3i(χ2i)= χ6i , σ5i(χ1i)= χ6i , σ7i(χ1i)= χ2i  and i=1,2.

By proposition (4.1) then θ4i= χ2i+ χ6i

2- (I) if i=1

θ61= χ21+ χ61

θ61([1,1])=2+2=4

θ61([1,r])= 2+2=4

θ61 ([x8,1])= 2+2=4

 θ61 ([x8,r])= 2+2=4

θ61 ([x,1])= V2+ V6=0

 θ61 ([x,r])= V2+ V6=0

θ61 ([x2,1])= 0+0=0

 θ61 ([x2,r])= 0+0=0

θ61 ([x4,1])= (-2)+ (-2)= -4

θ61 ([x4,r])= (-2)+ (-2)= -4

θ61 ([y,1])=0

θ61([y, r])=0

θ61 ([xy,1])=0

θ61([xy, r])=0

     

   2- (II) if i=2             

θ62= χ22+ χ62

 θ62 ([1,1])= 2+2=4

θ62 ([1,r])= (-2)+ (-2)= -4

θ62 ([x8,1])= 2+2=4

 θ62 ([x8,r])= (-2)+ (-2)= -4

θ62 ([x,1])= V2+ V6=0

 θ62 ([x,r])= (-V2)+(- V6)=0

θ62 ([x2,1])= 0+0=0

θ62 ([x2,r])= 0+0=0

θ62 ([x4,1])= (-2)+ (-2)= -4

θ62 ([x4,r])= 2+2=4

θ62 ([y,1])=0

θ62([y, r])=0

θ62 ([xy,1])=0

θ62([xy, r])=0

   The elements [x,1],[x3,1],[x5,1],[x7,1]

 are in the same  г-conjugate and[x, r],

   [x3, r], [x5, r],[x7, r] are in the same

г-conjugate and[x2,1],[x6,1] are in the same

 г-conjugate and [x2, r],[x6,r]are in the same

 г-conjugate as Table (6) .

Theorem (4.4)

  The rational valued characters table of the group Q2m×C2 when m=2h , h∈Z+ is given as follows:

      ≡*(Q2m×C2) = ≡*(Q2m)  ≡*(C2)

Proof:-

Since

                                                                        

1                                                                    1

1                                                                  -1

≡*(C2)=

From the definition of Q2m×C2, (theorem (3.4)),

                               ≡Q2m×C2 = (≡Q2m) (≡C2)

each element in Q2m×C2

      Q2m ,   C2      n=1,2,3,…,4m , s=1,2

and each irreducible character of Q2m×C2 is

                      

Where  is an irreducible character of Q2m and   is an irreducible character of C2 , then

From proposition (4.1)

                                                                                 

Where  is the rational valued of character table of Q2m×C2

Then ,

                                                                        

(I) if j=1 and s=1,2

 Where   is the rational valued character of Q2m.

(II) (a) if j=2 and s=1

       (b) if j=2 and s=2

                                                                                 

From (I) and (II) we have                                           

Then      ≡*(Q2m×C2) = ≡*(Q2m)  ≡*(C2)

The rational character table of the quaternion group (Q2m×C2)  when m=2h , h∈Z+ (4.5)

   From  Theorem (4.4) and form of ≡*(Q2h+1)=  in then table (5)  then  The rational character table of the quaternion group (Q2m×C2)  when m=2h , h∈Z+ is given in the general ( ) matrix form ≡*(Q2h+1×C2) as Table (7) .

Example (4.3)

   By using Theorem (4.4) the rational valued characters table of Q16×C2, Since h=3 ,it is same to the table Example (4.3), (after  change order the rows and the columns) as Table (8) .

References

 [1] C. W. Curtis & I. Reiner           ''Representation Theory of Finite Groups and Associative Algebras'', AMS Chelsea publishing, 1962, printed by the AMS, 2006.

[2] M. S. Kirdar '' The factor group of the Z-valued class function modulo the group of the Generalized characters'',Ph.D. thesis, University of Birmingham, 1982.

[3] N. R. Mahamood ''The Cyclic Decomposition of The Factor Group of (Q2m)/ (Q2m)''.M. Sc thesis, Technology University, 1995