International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 1 Artin's Characters Table of the Group (Q2n×D3) When n= , and are Primes Number Lecturer Naba Hasoon Jaber University of Krufa, College of Education for Girls, Department of Mathematics, Iraq Email:nabaah.al-saedi@uokufa.edu.iq Abstract: The main purpose of this paper is to find Artin's characters table of the group (Q2n×D3)when n= ,and are primes number, which is denoted by Ar(Q2n×D3) where Q2m is denoted to Quaternion group and D3 is the Dihedral group of order 6 . Keywords: Artin, characters,group, Q2n , D3 ,prime. 1. INTRODUCTION Let G be a finite group, two elements of G are said to be Г-conjugate if the cyclic subgroups they generate are conjugate in G and this defines an equivalence relation on G and its classes are called Г-classes [3]. Let H be a subgroup of G and let be a class function on H, the induced class function on G, is given by: ′(g) = ∑ , g where is defined by: (x) = { [2]. Let H be a subgroup of G and be a character of H, then   is a character of G, and it is called the induced character on G[7]. In 1976 ,David.G[3] studied "Artin Exponent of arbitrary characters of cyclic subgroup ", Journal of Algebra,61,p 58-76. In 1996, Knwabusz .K[9] studied "Some Definitions of Artin's Exponent of finite Group", USA, National foundation Math,GR. In this work we find Artin's characters table of the qroup (Q2n×D3) when n= ,and are primes number. 2. PRELIMINARIES 2.1The Generalized Quaternion Group Q2m [7 ] For each positive integer m,the generalized Quaternion Group Q2m of order 4m with two generators x and y satisfies Q2n={x h yk ,0 ,k=0,1} which has the following properties {x2n=y4=I, yxny-1=x-n}. 2.2The Dihedral Group D3 [9 ] The Dihedrael Group D3 is generate by a rotation r of order 3 and reflection s of order 2 then 6 elements of D3 can be written as: {1,r,r2,s,sr,sr2} . 2.3The Rational valued characters table: Definition(2.3.1) [5 ] A rational valued character θ of G is a character whose values are in Z,which is θ(g) Z for all g G. Theorem (2.3.2)[9 ] Every rational valued character of G be written as a linear combination of Artin's characters with coefficient rational numbers. Corollary (2.3.3)[5 ] The rational valued characters ∑ Form a basis for  ,where are the irreducible characters of G and their numbers are equal to the number of conjugacy classes of a cyclic subgroup of G. Proposition(2.3.4)[9 ] The number of all rational valued characters of finite G is equal to the number of all distinct Γ-classis. Definition (2.3.5)[5 ] The information about rational valued characters of a finite group G is displayed in a table called a rational valued characters table of G.We denote it by ≡⃰(G)which is matrix whose columns are Γ-classes and rows are the valuses of all rational valued characters where is the number of Γ-classes. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 2 The rational character table of Q2m where m is an odd number( 2.3.6) [7 ] Table(1) Where 0≤r≤ɭ ,ɭ is the number of Γ-classes C2m ,θj such that 1≤j≤ɭ+1 are the rational valued characters of group Q2m and if we denote Cij the elements of ≡⃰(Cm) and hij the elements of H as defined by: Hij={ The rational character table of Q2n when n= ,and are primes number (2.3.7)[7] Table(2) The rational character table of D3(2.3.8)[6] ≡⃰(D3) Table(3) 3. ARTIN'S CHARACTER TABLES: Theorem(3.1):[5] Let H be a cyclic subgroup of G and h1,h2,...,hm are chosen representatives for the m-conjugate classes of H contained in CL(g) in G,then: [y] Γ-classes of c2m X2r+1 X2r 1 1 1 1 ≡⃰(Cm) 1 1 1 ≡⃰(Cm) 0 0 -1 1 1 1 H 1 1 1 ≡⃰(Cm) 0 0 0 -2 -2 -2 2 2 2 [y] [x] [xn] [x2] [1] Γ-classes 1 1 1 1 1 0 -1 n-1 -1 n-1 1 1 1 1 1 0 1 1-n -1 n-1 0 -2 -2 2 2 [s] [r] [I] classes -Γ 3 2 1 2 3 6 0 -1 2 1 1 1 1 1 1 International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 3  (g)={ ∑ Propostion(3.2)[5 ] The number of all distinct Artin's characters on a group G is equal to the number of Γ-classes on G.Furthermore , Artin's characters are constant on each Γ-classes. Theorem(3.3) [10 ] The Artin's characters table of the Quaternion group Q2n when m is odd number is given as follows: Table(4) Where 0≤r≤m-1 , ɭ is the number of Γ-classes of C2m and are the Artin characters of the Quaternion group Q2m,for all 1≤j≤l+1. Artin characters table of Q2n when n= ,and are primes number (4.4)[ 8] The general form of Artin's characters of Q2n when n= ,and are primes number Table(5) The Artin characters of D3 (4.5)[9 ] Table(6) Γ-classes of C2m Γ-classes X2r X2r+1 [y] | 1 2 2 1 2 2n | 4n 2n 2n 4n 2n 2 2Ar(C2n) 0 0 0 m 0 0 n 0 0 1 Γ-classes [1] [x2] [xn] [x] [y] l l 1 2 1 2 2n l l 4n 2n 4n 2n 2 4n 0 0 0 0 4 4 0 0 0 2n 0 2n 0 0 2 2 2 2 0 n 0 n 0 1 Γ-classes [I] [r] [s] l l 1 2 3 ( )l 6 3 2 6 0 0 2 2 0 3 0 1 International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 4 4. THE MAIN RESULTE Propostion(4.1) If n = ,and are primes number, then The Artin's character table of the group (Q2n D3) is given as: The general form of the Artin characters of the group(Q2n D3)when n= ,and are primes number Table(7) which is (5 5) square matrix . Proof: Let g 2nxD3); g=(q,d),q Q2n,d D3 Case(I): If H is a cyclic subgroup of (Q2n {I}),then 1H= (x,I) 2H= (y,I) And the principle character of H, Artin's characters of Q2n,1≤j≤l+1, then by using theorem (4.1) { ∑ } 1H= (i) If g=(1,I) (j,1)(1,I)= . (g)= = = =6. (1) since H (ii) If g=(x n ,I),g then (j,1)(g)= = = =6. ( ) since H (iii) If g=(x 2 ,I) or g=(x,I) and g then (j,1)(g)= | | = (1+1)= .2= =6. ( since H and (g)=ø( )=1 and since g=(q,I),q Q2n,q x n (iv) if g H then Γ-classes [1,I][x2,I][xn,I][x,I][y,I] [1,r][x2,r][xn,r][x,r][y,r] [1,s][x2,s][xn,s][x,s][y,s] 1 2 1 2 2n 1 2 1 2 2n 1 2 1 2 2n 24n 24n 12n 12 24n 12n 24n 12n 12 24p 12n 24n 12n 12 6Ar(Q2n) 0 0 2Ar(Q2n) 2Ar(Q2n) 0 3Ar(Q2n) 0 Ar(Q2n) International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 5 (j,1)(g)=0 since H CL(g)= 2If H=(y,I)={(1,I),(y,I)(y2,I)(y3,I)} then (i) If g=(1,I) then (l+1,1)(g)= = =6.n=6. l+1(1) since H CL(1,I)={(1,I)} (ii) If g=(x n ,I)=(y 2 ,I) and g then (l+1,1)(g)= = =6.n=6. l+1(x n) since H CL(g)={g}, (g)=1 (iii) If g (x n ,I) and g H ,i.e.{g=(y,I) or g=(y 3 ,I)} then (l+1,1)(g)= -1))= =3.2=6. l+1(y)since H CL(g)={g,g -1} and (g)= (g-1)=1 Otherwise (l+1,I)(g)=0 since H CL(g)= Case(II): If H is a cyclic subgroup of (Q2nx{r}) then: 1H=(x,r) 2H=(y,r) 1-H=(x,r) and the principle character of H, then by using theorem (4.1) { ∑ } (i) If g=(1,I),(1,r) then (j,2)(g)= = = = =2. j(1) since H CL(g)={(1,I),(1,r),(1,r2)} (ii) g=(1,I),(xn,I),(xn,r),(1,r); g if g=(1,I),(1,r) then (j,2)(g)= (g)= since H CL(g)={g} and (g)=1 = = =2 j(1) (iii) if g= (xn,I),(xn,r) then (j,2)(g)= (g)= = = =2 j( ) (iv) if g (xn,I),(xn,r) and g then (j,2)(g)= (g)+ (g-1))= since H CL(g)={g,g-1} and (g)= (g-1)=1 = = Since g=(q,r),q Q2n,q x n (v) then (j,2)(g)=0 = j(q) since H CL(g)= 2if H=(y,r)={(1,I),(y,I),(y2,I),(y3,I),(1,r),(y,r),(y2,r),(y3,r)} (i) if g=(1,I),(1,r) then (l+1,2)(g)= = =2n=2 l+1(g) (ii) if g=(y2,I)=(xn,I),(y2,r) and g then International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 6 (l+1,2)(g)= = =2n=2 l+1(g) since H CL(g)={g} and (g)=1 (iii) if g (x n ,I) and g i.e. g={(y,I),(y,r)} or g={(y 3 ,I),(y 3 ,r)} then (l+1,2)(g)= (g)+ (g-1))= = 2 l+1(g) since H CL(g)={g,g-1} and (g)= (g-1)=1 otherwise (l+1,2)(g)=0 since H CL(g)= case(III): if H is a cyclic subgroup of (Q2nx{s}) then 1H=(x,s), 2H=(y,s) and the principle character of H, then by using theorem (4.1) { ∑ } 1H=(x,s) (i) If g=(1,I) then (j,3)(g)= | | (g)= = since H CL(g)={(1,I)} If g={(1,s)} then (j,3)(g)= | | (g)= = since H CL(g)={(1,s)} (ii) If g=(1,I),(xn,I),(xn,s),(1,s); g then If g=(1,I) then (j,3)(g)= (g)= since H CL(g)={g} and (g)=1 = = =3 j(1) If g={(1,s)} then (j,3)(g)= | | (g)= = since H CL(g)={g} and (g)=1 (iii)If g= (xn,I) then (j,3)(g)= (g)= = = =3 j( ) If g= (xn,s) then (j,3)(g)= (g)= = = = j( ) (iv)If g (xn,I),(xn,s) and g If g (xp,I) then (j,3)(g)= (g)+ (g-1)) = since H CL(g)={g,g-1} and (g)= (g-1)=1 = = j Since g=(q,I),q Q2n , q x n If g (xn,s) then (j,3)(g)= (g)+ (g-1)) = since H CL(g)={g,g-1} and (g)= (g-1)=1 = = j Since g=(q,s),q Q2n , q x n then (j,3)(g)=0 = j(q) since H CL(g)= International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 4, April – 2019, Pages: 1-8 www.ijeais.org 7 2-if H=(y,s)={(1,I),(y,I),(y2,I),(y3,I),(1,s),(y,s),(y2,s),(y3,s)} then (i)If g=(1,I) then (l+1,3)(g)= = =3.n=3 l+1(g) If g=(1,s) then (l+1,3)(g)= = =n= l+1(g) (ii)If g=(y2,I)=(xn,I) and g then (l+1,3)(g)= = = 3.n=3 l+1(g) since H CL(g)={g} and (g)=1 If g=(y2,s) and g then (l+1,3)(g)= = =n= l+1(g) since H CL(g)={g} and (g)=1 (iii)If g (xn,I) and g i.e. g={(y,I),(y,s)} or g={(y3,I),(y3,s)} then (l+1,3)(g)= (g)+ (g-1))= = 3 l+1(g) since H CL(g)={g,g -1 } and (g)= (g -1 )=1 (iv)If g=(y2,s), g H then (l+1,3)(g)= (g)= = = l+1(g) (v)If g=(y,s) then (l+1,3)(g)= (g)+ (g-1))= = = since H CL(g)={g,g-1} and (g)= (g-1)=1 otherwise (l+1,3)(g)=0 sinceH CL(g)= Example (4.2): To find Artine's character table of the group (Q66xD3) . Ar(Q66xD3)= Ar(Q2.3.11xD3)= Table(8 ) Γ-classes [1,I] [x2,I] [x33,I] [x,I] [y,I] [1,r] [x2,r] [x33,r] [x,r] [y,r] [1,s] [x2,s] [x33,s] [x,s] [y,s] | | 1 2 1 2 2n 2 2 2 2 2n 3 3 3 3 6n | 792 396 792 396 12 396 396 396 396 12 264 264 264 264 4 (1,1) 792 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2,1) 264 264 0 0 0 0 0 0 0 0 0 0 0 0 0 (3,1) 396 0 132 0 0 0 0 0 0 0 0 0 0 0 0 (4,1) 24 0 0 24 0 0 0 0 0 0 0 0 0 0 0 (5,1) 8 8 0 8 8 0 0 0 0 0 0 0 0 0 0 (1,2) 12 0 4 12 0 4 0 0 0 0 0 0 0 0 0 (2,2) 396 0 0 0 0 0 396 0 0 0 0 0 0 0 0 (3,2) 132 132 0 0 0 0 132 132 0 0 0 0 0 0 0 (4,2) 198 0 66 0 0 0 198 0 66 0 0 0 0 0 0 (5,2) 12 0 0 12 0 0 12 0 0 12 0 0 0 0 0 (1,3) 4 4 0 4 4 0 4 4 0 4 4 0 0 0 0 (2,3) 6 0 2 6 0 2 6 0 2 6 0 2 0 0 0 (3,3) 198 0 0 0 0 0 198 0 0 0 0 0 6 0 0 (4,3) 66 66 0 0 0 0 66 66 0 0 0 0 2 2 0 (5,3) 99 0 33 0 0 0 99 0 33 0 0 0 3 0 1 REFERENCES: [1] M.J.Hall "The Theory of Group ",Macmillan,Neyork,1959. 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