RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS

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1 Jounal of Algebaic Systems Vol. 3, No. 2, (2016), pp RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS H. SHABANI, A. R. ASHRAFI AND M. GHORBANI Abstact. The aim of this wok is to compute the ational chaacte tables of the dicyclic goup T 4n, the goups of the odes pq and pq. Some geneal popeties of ational chaacte tables ae also consideed. 1. Intoduction In this section, we establish some basic notation, and teminologies that ae used thoughout this aticle. Let p be a pime numbe, and q be a positive intege such that q p 1. Define the goup F p,q to be pesented by F p,q = a, b : a p = b q = 1, b 1 ab = a u, whee u is an element of ode q in the multiplicative goup Z p [5, Page 290]. In the case that q is also a pime, we use the notation T p,q as F p,q. It is easy to see that F p,q is a Fobenius goup of the ode pq. Hölde [3] classified goups of ode pq; p, q, and ae the pimes. Using his esults, we can pove that all goups of the ode pq (p > q > ) ae isomophic to one of the following goups: G 1 = Z pq, G 2 = Z F p,q (q p 1), G 3 = Z q F p, ( p 1), G 4 = Z p F q, ( q 1), G 5 = F p,q (q p 1), MSC(2010): Pimay: 20C15; Seconday: 20D15 Keywods: Rational chaacte table, Chaacte table, Galois goup. Received: 8 Januay 2015, Revised: 11 Januay Coesponding autho. 151

2 152 SHABANI, ASHRAFI, AND GHORBANI G i+5 = a, b, c : a p = b q = c = 1, ab = ba, c 1 bc = b u, c 1 ac = a vi, whee p 1, q 1, o(u) = in Z q and o(v) = in Z p (1 i 1). Suppose that G is a finite goup, I(G) denotes the set of all ieducible chaactes of G, Cl(G) is the set of conjugacy classes of G, and x, y G. Also assume that A is the set of all chaacte values of G, Q(A) denotes the field geneated by Q and A. If Γ is the Galois goup of this extension, then Γ acts on I(G) by χ α (g) = α(χ(g)). It is well-known that thee exists ε such that Q(A) Q(ε), whee ϵ is a pimitive n th oot of unity. Thus if α Γ, then thee exists a unique positive intege such that (, n) = 1 and α(ε) = ε. Theefoe, it is well-defined to use the notation α = σ. The elements x and y ae said to be ational conjugates, if x and y ae the conjugate subgoups of G. The obits of G unde this action ae called the ational conjugacy classes of G. On the othe hand, Γ acts on the conjugacy classes of G by (x G ) σ = (x ) G. It is well known that [4, Coollay 6.33] the numbe of obits in the actions of Γ on the ieducible chaactes and conjugacy classes of G ae equal. Moeove, the obits of Γ on the conjugacy classes of G ae the ational conjugacy classes of G. Lemma 1.1. Suppose that G is a finite goup, A is the set of all chaacte values of G, and Γ = Gal( Q(A) ). Then the following hold: Q (1) If O is an obit of Γ unde its action on conjugacy classes of G, then the union of elements of O is a ational conjugacy class of G, and each ational conjugacy class of G can be obtained in this way. (2) If O(χ) denotes the obit of χ unde the action of Γ on I(G), then the summation of all ieducible chaactes of O(χ) is a ational-valued chaacte of G. Moeove, fo any pope subset T of O(χ), the summation of all ieducible chaactes of T is not ational-valued. Poof. Suppose that A = {χ(g) χ I(G) & g G}. Then: (1) Clealy, fo each one of the conjugacy class x G and y G of an obit O, thee exists σ Γ such that (, G ) = 1 and σ (x G ) = (x ) G = y G. Theefoe, thee exists g G such that y = g 1 x g. This implies that x and y ae ational conjugates. (2) The sum ove O(χ) unde the Galois goup of the field Q(A), say, which is geneated by the values of χ, is ational because the value of the obit sum at the goup element g is equal to

3 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 153 the tace of χ(g) with espect to the field extension Q(A)/Q multiplied by the ode of the stabilize of χ(g) in Gal(Q(A)/Q). Suppose that the sum ove a pope subset T of O(χ) would be ational-valued, and assume, without loss of geneality, that χ is in T. Take an element α in Gal(Q(A)/Q) that maps χ to a chaacte in the complement of T in O(χ). Then the sum ove the set {ψ α ψ T } is equal to the sum ove T. This contadicts the linea independence of the chaactes in O(χ). This ends the poof. Suppose that X = {X 1,, X } denotes the set of obits of Γ on I(G), and K = {K 1,, K } is the set of obits of Γ on Cl(G). Then the chaactes γ i = m i ψ X i ψ and 1 i ae called the ieducible ational chaactes of G, whee m i denotes the Schu index of a chaacte in X i. Definition 1.2. Let the pai (K, X ) be as defined above. The squae matix QCT (G) = [a ij ] with a ij = γ i (K j ) is said to be the ational chaacte table of G, whee γ i (K j ) is equal to γ i (g) fo any g K j. Thoughout this pape, ou notations ae standad, and can be taken fom [2]. Ou calculations ae caied out with the aid of the compute algeba system GAP [6]. 2. Main esults Suppose that G is a goup, and A is the set of all chaacte values of G. In the fist section, it is shown that thee ae two actions of Γ = Gal( Q(A) ) on I(G) and Cl(G). Suppose that G is a finite goup Q with QCT (G) = (γ i (K j )), whee 1 i, j, and denotes the numbe of ieducible ational chaactes. Then the ow and column othogonality elations fo the ational chaacte table of G can be witten in the following fom: γ i, γ j = 1 K t γ i (K t )γ j (K t ) = δ i,j m 2 i K i, G t=1 K p, K q = γ i (K p )γ i (K q ) = δ p,q m 2 p C G (K p ) K p. i=1 Thus if g and g ae two ational conjugate elements, then C G (g) = C G (g ). In the above elations, K i is equal to the numbe of conjugacy classes g G such that K i = g G.

4 154 SHABANI, ASHRAFI, AND GHORBANI The following poposition extends a well-known esult of the odinay chaacte theoy to that fo the ational chaacte theoy [5, Page 167]. Poposition 2.1. Let G be a finite goup. Then, det(qct (G)) 2 = ( C G (K i ) K i ). i=1 Poof. Suppose Q = QCT (G) = (γ i (K j )). Then, Q t Q = (γ j (K i )) (γ i (K j )) = ( γ i (K s )γ i (K t )) i=1 = diag( C G (K i ) K i ). Hence, det(qct (G)) 2 = i=1 ( C G(K i ) K i ), poving the esult. Poposition 2.2. If G is cyclic, then the identity matix. QCT (G)2 G = I, whee I denotes Poof. Suppose (K, X ) is as definition 1 fo G such that K = {K 1,, K }. Conside the following two matices and A = diag( K 1,, K ), B = 1 1 diag( G K1,, 1 K ). Set C = BQA, whee Q = QCT (G). Then CC t = I, and since G is cyclic, CC t = Q2 = I, as desied. G Let µ(n), ϕ(n), and τ(n) denote the Möbius µ-function, Eule totient function, and numbe of diviso of n, espectively. Fo n, m Z with n 1, the Ramanujan sum c n (m) is defined as c n (m) = 1 k n (k,n)=1 e 2πikm n. The Ramanujan sum is always an intege, and the von Stenecks Fomula says that c n (m) = µ ( (n,m))ϕ(n) n. ϕ( n (n,m)) Theoem 2.3. Fo a positive intege n, let d 1, d 2,..., d τ(n) be the divisos of n. The ational chaacte table of the cyclic ( goups Z n can n be computed using QCT (Z n ) = [a ij ], whee a ij = c di d j ).

5 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 155 Poof. Suppose that τ(n) denotes the numbe of divisos of n. The ieducible chaactes of the cyclic goup Z n can be computed by χ k (m) = ε km, whee ε = e 2πi n, and the Schu index of each chaacte is equal to 1. It is well-known that Q(A) = Q(ε), and Γ = Gal( Q(A) ) Q = Z n. Then X = {X 1,, X τ(n) } and K = {K 1,, K τ(n) }, whee X i = {χ k (k, n) = n d i } and K j = {a Z n (a, n) = n d j }. This computes the ational chaacte table of Z n, as desied. Example 2.4. Let p be a pime numbe. The ational chaacte table of Z p is ecoded in Table 1. In this case, K 1 = id Zp and K 2 = Z p id Zp. Table 1. Rational Chaacte Table of Z p. QCT (Z p ) K 1 K 2 γ γ 2 p 1 1 Example 2.5. Following James and Liebeck [5, p. 178], the dicyclic goup T 4n can be pesented as follows: T 4n = a, b a 2n = 1, a n = b 2, b 1 ab = a 1. It is easy to see that this goup has the ode 4n and the cyclic subgoup a has the index 2 in T 4n. Suppose that d 1,, d τ(n) ae the positive divisos of n. Let ε = e 2πi 2n and I( a ) = {ψ 1, ψ 2,, ψ 2n }, whee ψ j (a ) = ε j, and = 1, 2,, 2n. In [5, p. 420], the chaacte table of T 4n is computed. Fo the sake of completeness, we descibe the chaacte table of T 4n as follow: The goup T 4n has exactly n + 3 conjugacy classes as follows: {e}, {a n }, {a, a } (1 n 1), {a 2j b 0 j n 1}, and {a 2j+1 b 0 j n 1}. The goup T 4n has exactly n 1 non-linea chaactes ψ j such that ψ j a = ψ j + ψ n j, whee 1 j n 1. It is easy to see that ψ j (a α b β ) = (1 β)(ε jα + ε jα ). The Schu index of ψ j is equal to 2, whee j is odd and is equal to 1 othewise. The goup T 4n has exactly fou linea chaactes. These ae {ψ n, ψ n} and {χ 1, χ 2 } such that ψ n and ψ n ae two linea chaactes of T 4n with this popety that ψ n a = ψ n a = ψ n, χ 1 = 1 and χ 2 is the lift of the non-tivial ieducible chaacte of T 4n a = Z2 to T 4n. Notice that if n is odd, then

6 156 SHABANI, ASHRAFI, AND GHORBANI ψ n (a α b β ) = ( 1) α i β and ψ n(a α b β ) = ( 1) α ( i) β. If n is even, ψ n (a α b β ) = ( 1) α ( 1) β and ψ n(a α b β ) = ( 1) β. It is obvious the Schu index of these chaactes is equal to 1. On the othe hand, Q(A) = Q(ε + ε 1 ), whee A is the chaacte values of T 4n and [Q(A) : Q] = ϕ(2n). Thus the Galois goup is abelian 2 of ode ϕ(2n). Suppose that n is odd. Then the action of Galois goup 2 Γ on I(T 4n ) has exactly τ(2n) + 1 obits, as follow: X 1 = {1}, X 2 = {χ 2 }, X i = {ψ j (j, 2n) = 2n ; j = 1,, n 1} (3 i τ(2n)), d i X τ(2n)+1 = {ψ n, ψ n}. Notice that X i = ϕ(d i) 2. We now conside the action of Galois goup Γ on Cl(T 4n ). The obits of this action ae as follows: K 1 = {1}, K 2 = {a n }, K i = {a, a 2n (, 2n) = 2n } (3 i τ(2n)), d i K τ(2n)+1 = {a j b 0 j 2n 1}. Then X = {X 1, X 2,, X τ(2n), X τ(2n)+1 } and K = {K 1, K 2,, K τ(2n), K τ(2n)+1 }. We now assume that n is even. Then, simila to the case of odd n, we have exactly τ(2n)+2 obits, as follow: X 1 = {1}, X 2 = {χ 2 }, X i = {ψ j (j, 2n) = 2n ; j = 1,, n 1} (3 i τ(2n)), d i X τ(2n)+1 = {ψ n }, X τ(2n)+2 = {ψ n}.

7 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 157 Again X i = ϕ(d i) 2, and we have: K 1 = {1}, K 2 = {a n }, K i = {a, a 2n (, 2n) = 2n } (3 i τ(2n)), d i K τ(2n)+1 = {a 2j b 0 j n 1}, K τ(2n)+2 = {a 2j+1 b 0 j n 1}. Theefoe, we have: X = {X 1, X 2,, X τ(2n)+1, X τ(2n)+2 }, K = {K 1, K 2,, K τ(2n)+1, K τ(2n)+2 }. Notice that, by [7], the Schu index of T 4n can be computed as follows: { 1 Xi contains ψj when j is even t i = 2 X i contains ψj when j is odd. The ational chaacte tables of T 4n ae ecoded in Tables 2 and 3. Table 2. Rational Chaacte Table of T 4n ; n is odd. QCT (T 4n ) K 1 K 2 K i K τ(2n)+1 γ γ γ i t i ϕ(d i ) t i c di (n) t i c di () 0 γ τ(2n) ( 1) 0 Table 3. Rational Chaacte Table of T 4n, n is even. QCT (T 4n ) K 1 K 2 K i K τ(2n)+1 K τ(2n)+2 γ γ γ i t i ϕ(d i ) t i c di (n) t i c di () 0 0 γ τ(2n) ( 1) 1 1 γ τ(2n) ( 1) 1 1 Lemma 2.6. Let A, A 1, and A 2 be the set of all enties of the chaacte tables G 1 G 2, G 1 and G 2, espectively. Let Γ 1 = Gal( Q(A 1) ), Γ Q 2 = Gal( Q(A 2) Q Q(A) ) and Γ = Gal( ). If gcd( G Q 1, G 2 ) = 1, then Γ = Γ 1 Γ 2.

8 158 SHABANI, ASHRAFI, AND GHORBANI Poof. The poof follows fom the fact that all enties of the chaacte tables G 1 and G 2 ae the sum of oots of unity of the odes G 1 and G 2, espectively. Lemma 2.7. [4, Execise 10.10] Let χ I(G 1 ), ψ I(G 2 ), and gcd(m(χ), ψ(1) Q(ψ) : Q ) = gcd(m(ψ), χ(1) Q(χ) : Q ) = 1. Then m(χψ) = m(χ)m(ψ). Theoem 2.8. Suppose that G 1 and G 2 ae finite goups of the odes n 1 and n 2, espectively, whee gcd(n 1, n 2 ) = 1. If χ and ψ satisfy the hypothesis of Lemma 2.7 fo any χ I(G 1 ) and ψ I(G 2 ), then we have QCT (G 1 G 2 ) = QCT (G 1 ) QCT (G 2 ). Poof. By Lemma 2.6, Γ = Γ 1 Γ 2, whee Γ 1 = Gal( Q(A 1) ), Γ Q 2 = Gal( Q(A 2) ), and Γ = Gal( Q(A) ). Let A, A Q Q 1 and A 2 be the enties of the chaacte tables G 1 G 2, G 1, and G 2, espectively. Suppose that the action of Γ 1 on I(G 1 ) and the action of Γ 2 on I(G 2 ) have the classes {X 1,..., X } and {Y 1,..., Y s }, espectively. It is well-known that, I(G 1 G 2 ) = I(G 1 ) I(G 2 ). Thus it is sufficient to pove that fo each χ X i, (1 i ) and fo each ψ Y j, (1 j s), [χψ] = [χ][ψ], whee [χ] = χ σ. It is clea that fo evey σ Γ, σ Γ thee exist σ 1 Γ 1 and σ 2 Γ 2 such that σ = (σ 1, σ 2 ). This implies that [χψ] = {(χψ) σ σ Γ} = {(χψ) (σ 1,σ 2 ) σ 1 Γ 1, σ 2 Γ 2 } = {χ σ 1 ψ σ 2 σ 1 Γ 1, σ 2 Γ 2 } = [χ][ψ]. Suppose that X i = {χ it } t and Y j = {ψ jl } l, then if γ i = m i t χ i t and δ j = m j l ψ j l, by Lemma 2.7, we have: γ i δ j = m i m j χ it ψ jl. Similaly, the ational conjugacy classes of G 1 G 2 ae equal to the poduct of the ational conjugacy classes of G 1 and G 2. Let K f and L t be the ational conjugacy classes of G 1 and G 2, espectively. Then we have γ χi ψ j ((K f, L s )) = γ χi (K f )γ ψj (L s ). This completes the poof. t,l

9 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 159 Example 2.9. Let p and q be two pime numbes such that p > q, and G be a finite goup of the ode pq. It is well-known that G = Z pq o G = T p,q. As a esult of Theoem 2.8, one can easily pove that the ational chaacte table of Z p Z q is QCT (Z p ) QCT (Z q ). Table 4. Rational Chaacte Table of Z pq. QCT (Z pq ) K 1 K 2 K 3 K 4 γ γ 2 q 1 1 q 1 1 γ 3 p 1 p γ 4 pq p q p 1 q 1 Hee, K 1 = {id}, K 2 = {p, 2p,..., (q 1)p}, K 3 = {q, 2q,..., (p 1)q}, and K 4 = a Z pq : (a, pq) = 1}. Poposition Suppose that p is pime, and q p 1 and d 1,, d τ(q) ae the positive divisos of q. Then the ational chaacte table of G = F p,q is given in Table 5. Table 5. Rational Chaacte Table of F p,q. QCT (F p,q ) K 1 K 2 K j γ γ i ϕ(d i ) ϕ(d i ) c di ( q d j ) γ τ(q)+1 p and β = e 2πi p. Accoding to [5, Poposi- Poof. Suppose that α = e 2πi q tion 25.9 and Theoem 25.10], the q linea chaactes of F p,q ae defined non- as χ i (a x b y ) = α iy, whee 0 i q 1. Moeove, the = p 1 q linea chaactes of F p,q ae ψ j (g) = { s S βsxv j g = a x 0 g = a x b y, whee S is a subgoup of Z p containing all powes of u, and the epesentatives of the cosets of S in Z p can be consideed as {v 1, v 2,..., v }. The conjugacy classes of F p,q ae: {1}, (a v i ) G = {a v is : s S}, (1 i ), (b n ) G = {a m b n : 0 m p 1}, (1 n q 1).

10 160 SHABANI, ASHRAFI, AND GHORBANI Then the action of the Galois goup Γ = Gal( Q(α, s S βs ) Q ) on I(F p,q ) has exactly τ(q) + 1 obits, as: X 1 = {1}, X i = {χ j (j, q) = q d i ; j = 1,, τ(q)} (2 i τ(q)), X τ(q)+1 = {ψ 1, ψ 2,..., ψ }. On the othe hand, the obits of the action of Galois goup Γ on Cl(F p,q ) ae: K 1 = {1}, K 2 = {a i : 1 i p 1}, K j+1 = (k,q)= q d j (b k ) G, (2 j τ(q)). By [1, Theoem 4.2], the Schu index of all ieducible chaactes of F p,q is 1, and the poof is completed. Coollay Let p and q be two pime numbes such that p > q and q p 1. The ational chaacte table of the goup T p,q is as Table 6. In this case, the ational conjugacy classes ae K 1 = {id}, K 2 = a \ {id} and K 3 = T p,q \ a. Table 6. Rational Chaacte Table of T p,q. QCT (T p,q ) K 1 K 2 K 3 γ γ 2 q 1 q 1 1 γ 3 p Fom now on, we conside the goups of the ode pq (p > q > ). To compute the ational chaacte tables of a goup G of this ode, we fist calculate the chaacte table of all goups of this ode. In Theoem 2.3, the ational chaacte table of the cyclic Z n is computed. Thus it is enough to assume that goup G is non-cyclic. We now apply Theoems 2.3 and 2.8 to compute the ational chaacte tables of Z F p,q, Z q T p, and Z p T q, as QCT (Z ) QCT (F p,q ), QCT (Z q ) QCT (T p, ), and QCT (Z p ) QCT (T q, ), espectively. The chaacte table of G 5 can be computed diectly fom Poposition We now compute the chaacte table of goups G i+5. Let G =

11 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 161 G i+5, to compute the conjugacy classes of G. We assume that U = u and V = v ae the subgoups of ode of Z q and Z p, espectively. Then we have, Lemma The conjugacy classes of G ae {1}, (a v i ) G, (b u i ) G, (c i ) G, (b u i a v i ) G whee u i is a coset epesentative of U in Z q, v i is a coset epesentative of V in Z p, and (u i, v i) is a coset epesentative of (u, v) in Z q Z p. Poof. It is easy to see that fo 1 k 1, c k bc k = b uk and so b ui s ae conjugate and so b G. On the othe hand, ba C G (b), and hence, C G (b) pq. This implies that b G, and thus b G =. Futhe, one can pove that (b u i ) G (1 i q 1 ) and (av j ) G (1 j ) ae conjugacy classes of G. Theefoe, p 1 c G = {cb i a j 0 i q 1, 0 j p 1},. (c 1 ) G = {c 1 b i a j 0 i q 1, 0 j p 1}, (b u i a v i ) G = {b u i a v i, b u i u a v i v,, b u i u 1 a v i v 1 }, whee (u i, v i) is a coset epesentative of (u, v) in Z q Z p (u, v) =. and It follows fom Lemma 2.12 that G has p 1 + conjugacy classes and then the same numbe of ieducible chaactes. On the othe hand, G/G = c c = 1 = Z. + q 1 + (p 1)(q 1) Hence, G has linea chaactes lifted fom linea chaactes of G/G. These chaactes ae as χ n : G/G C with χ n (c m G ) = ϵ mn, whee ϵ = e 2πi and m, n = 0, 1,, 1. Accoding to [5, Theoem 17.11], all linea chaactes of G ae as χ n : G C with χ n (g) = χ n (gg ). Hence, χ n (a w ) = χ n (a w G ) = χ n (G ) = χ n (1) = 1, χ n (b v ) = χ n (b v G ) = χ n (G ) = χ n (1) = 1, χ n (b v 0 a w 0 ) = χ n (b v 0 a w 0 G ) = χ n (G ) = χ n (1) = 1, χ n (c t ) = χ n (c t G ) = ϵ tn (0 n 1 and 1 t 1). whee (v 0, w 0 ) is a coset epesentative of (u, v) in Z q Z p.

12 162 SHABANI, ASHRAFI, AND GHORBANI Hee, we detemine all the non-linea ieducible chaactes of G. Fist notice that H = a is a nomal subgoup of G, and if u 1(mod q), then G/H = b, c b q = c = 1, c 1 bc = b u = F q,. Accoding to Poposition 2.10, the Fobenius goup F q, has linea chaactes and q 1 ieducible chaactes of degee. Let us denote the non-linea chaactes by φ m. Then we have: φ m (H) =, φ m (b x H) = 1 λ u mxu i (1 m q 1, 1 x q 1), i=0 φ m (b x c y H) = 0 (1 y 1), whee λ = e 2πi q and u 1,, u m ae distinct coset epesentative of U = u in Z q. By lifting these chaactes, we can compute q 1 ieducible chaactes of G of degee denoted by φ m (1 m q 1 ), e.g. φ m (a x ) = (0 x p 1), φ m (b y a x ) = 1 λ u myu i (0 x p 1, 1 y q 1), i=0 φ m (c k ) = 0 (1 k 1). Similaly, fo the nomal subgoup K = b of G, we have: G/K = a, c a p = c = 1, c 1 ac = a v = F p,. Consequently, this goup has p 1 ieducible chaactes of degee denoted by θ l (1 l p 1 ). Simila to the last discussion, the ieducible chaactes of G lifted fom θ l ae as follows: 1 θ l (a x ) = θ l (b y a x ) = γ v lxv i, θ l (b y ) =, i=0 θ l (c k ) = 0 (1 k 1), whee γ = e 2πi p and v 1,, v l ae distinct coset epesentative of V = v in Z p. Finally, by consideing subgoup G = ba = Z q Z p, its ieducible chaactes ae of the fom ψ i ξ j (0 i q 1, 0 j p 1) and ψ i (b y ) = λ iy, ξ j (a x ) = γ jx.

13 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 163 This leads us to conclude that Let now m Z q and n Z p, then On the othe hand, and so ψ i ξ j (b y a x ) = ψ i (b y )ξ j (a x ) = λ iy γ jx. (ψ m ξ n G)(1) = G ba (ψ mξ n )(1) = pq pq =. C G (b y ) = C G (b y ) = C G (a x ) = C G (a x ) = C G (b y a x ) = C G (b y a x ) = pq, (ψ m ξ n G)(a x ) = (ψ m ξ n G)(b y ) = (ψ m ξ n G)(b y a x ) = 1 1 ξ n (a xvi ) = γ nxvi, i=0 i=0 1 1 ψ m (b yui ) = λ myui, i=0 i=0 1 1 ψ m (b yui )ξ n (a xvi ) = λ myui γ nxvi i=0 (ψ m ξ n G)(c k ) = 0 (k = 1,, 1). Since ψ m ξ n G = ψ mu iξ nv i G, thus, we get z = (p 1)(q 1) chaactes of G. Thee still emains the question as to whethe such chaactes ae distinct ieducible. Assume (u i, v i) be coset epesentative of subgoup {(1, 1), (u, v),, (u 1, v 1 )} of Z q Z p and η j = ψ u ξ j v G. Accoding to Fobenius Recipocity Theoem, fo H = G = ba, we j veify: η j H, ψ u j ui ξ v Theefoe, we can obseve that j vj H = η j, ψ u j ξ v j = η j, η j G. i=0 i=0 G G 1 η j H = η j, η j G ( ψ v ξ j ui v ) + χ, j vi whee χ = 0 o it is a chaacte of H. Hence, η j (1) η j, η j G. Finally, η j (1) = implies that η j, η j G = 1, and so η j is ieducible. On the othe hand, fo (u j, v j) Z q Z p, all ψ u ξ j v s ae linealy independent, j and thus all η j H (1 j (p 1)(q 1) ) ae distinct. Consequently,

14 164 SHABANI, ASHRAFI, AND GHORBANI the ieducible chaactes η 1, η z ae distinct. We summaize the chaacte table of G in the following theoem: Theoem Let p > q > be distinct pime numbes, l 1 = (p 1)(q 1), l 2 = p 1, l 3 = q 1 and ϵ = e 2πi. Then the goup G has l 1 + l 2 + l 3 + ieducible chaactes, as epoted in Table 7. Table 7. Chaacte Table of Goup G i+5 fo 1 i. g 1 a v i b u i b u i a v i c k 1 i l 1 1 i l 2 1 i l 3 1 k 1 χ n ϵ kn 0 n 1 η s E F G 0 1 s l 3 θ l C D 0 1 l l 1 φ m A B 0 1 m l 2 whee λ = e 2πi q, u1,, u l1 ae distinct coset epesentative of U = u in Z q, v 1,, v l2 ae distinct coset epesentative of V = v in Z p, (u i, v i) ae coset epesentative of (u, v) in Z q Z p and A = λ u mu i u j, B = λ u mu i uj, C = γ v lv i v j, D = γ v lv i vj, E = j=1 γ v sv i v j, F = j=1 j=1 λ u su i u j, G = j=1 j=1 λ u su i uj γ v sv i vj. j=1 and 1 l l 1, 1 m l 2, 1 s l 3, 1 n 1. Suppose G(p, q, ) be the set of all goups of ode pq, whee p, q, and ae distinct pime numbes with p > q >. Theoem Let G G(p, q, ). Then, the ational chaacte table of G is equal to one of the Tables Poof. Using Theoems 2.3, 2.8 and Poposition 2.10, the ational chaacte table of goups G 1 G 5 ae as epoted in Tables Let G = G i+5 fo 1 i 1. Accoding to [7, Theoem 3], the Schu index of G is equal to 1. Since p, q, and ae pime numbes, one can easily check that a s = a t, b m = b n, c k = c l, and b x a y = b v a z, j=1

15 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 165 whee 1 s, t, y, z p 1, 1 m, n, x, v q 1, 1 k, l 1. This implies that all ational conjugacy classes ae K 1 = {id}, K 2 = l 1 i=1 (a v i ) G, K 3 = l 2 i=1 (b u i ) G, K 4 = l 3 i=1 (b u i a v i ) G, K 5 = 1 k=1 (ck ) G. On the othe hand, let Q(A) = Q(ϵ, λ u + λ u 1, γ v + γ v 1 ), so the action of Galois goup Γ on I(G) has exactly five obits, as follow: X 1 = {1}, X 2 = {χ n : 1 n 1}, X 3 = {η s : 1 s l 3 }, X 4 = {θ l : 1 l l 1 }, X 4 = {φ m : 1 m l 2 }. This completes the poof. 3. Appendix In this section, the ational chaacte tables of the goups G 1, G 2, G 3, G 4, G 5 and G i+5, 1 i 1, all of which having the ode pq ae pesented. These tables ae computed by the methods given by Poposition 2.10 and Theoems 2.3, 2.8, and Table 8. Rational Chaacte Table of Goup G 1. QCT (G 1 ) K 1 K 2 K 3 K 4 K 5 K 6 K 7 K 8 γ γ γ 3 q 1 q q 1 q γ 4 a 1 1 q 1 1 a 1 1 q 1 1 γ 5 p 1 p 1 p 1 p γ 6 a 2 1 p a 2 1 p γ 7 a 3 p q 1 q 1 1 γ 8 a 4 a 3 a 2 p 1 a 1 1 q 1 1 Hee, a 1 = (q 1)( 1), a 2 = (p 1)( 1), a 3 = (p 1)(q 1), and a 4 = (p 1)(q 1)( 1). The ational conjugacy classes ae,

16 166 SHABANI, ASHRAFI, AND GHORBANI K 1 = {id G1 }, K 2 = {pq, 2pq,..., ( 1)pq}, K 3 = {p, 2p,..., (q 1)p}, K 4 = {q, 2q,..., (p 1)q}, K 5 = {p, 2p,..., ( 1)(q 1)p}, K 6 = {q, 2q,..., ( 1)(p 1)q}, K 7 = {, 2,..., (q 1)(p 1)}, K 8 = {a Z pq : (a, pq) = 1}. Table 9. Rational Chaacte Table of Goup G 2. QCT (G 2 ) K 1 K 2 K 3 K 4 K 5 K 6 γ γ γ 3 q 1 q 1 q 1 q γ 4 a 1 1 q a 1 1 q 1 1 γ 5 p 1 p γ 6 a 2 1 p Hee, a 1 = (q 1)( 1) and a 2 = (p 1)( 1). Let F p,q = a, b : a p = b q = 1, b 1 ab = a u. Also conside that L 1 = {id Z } and L 2 = Z \ L 1 ae the ational conjugacy classes of Z and T 1 = {id Fp,q }, T 2 = {a,..., a p 1 } and T 3 = F p,q \ a ae the ational conjugacy classes of F p,q. Then we have K 1 = L 1 T 1, K 2 = L 2 T 1, K 3 = L 1 T 2, K 4 = L 2 T 2, K 5 = L 1 T 3, and K 6 = L 2 T 3. Table 10. Rational Chaacte Table of Goup G 3. QCT (G 3 ) K 1 K 2 K 3 K 4 K 5 K 6 γ γ 2 q 1 1 q 1 1 q 1 1 γ γ 4 a 1 1 a q 1 γ 5 p 1 p γ 6 a 2 1 p 1 q Hee, a 1 = (q 1)( 1) and a 2 = (p 1)(q 1). Let F p, = a, b : a p = b = 1, b 1 ab = a u. Conside L 1 = {id Zq } and L 2 = Z q \ L 1 ae the ational conjugacy classes of Z q and T 1 = {id Fp, }, T 2 = {a,..., a p 1 },

17 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS 167 and T 3 = F p, \ a ae the ational conjugacy classes of F p,. Then K 1 = L 1 T 1, K 2 = L 2 T 1, K 3 = L 1 T 2, K 4 = L 2 T 2, K 5 = L 1 T 3, and K 6 = L 2 T 3. Table 11. Rational Chaacte Table of Goup G 4. QCT (G 4 ) K 1 K 2 K 3 K 4 K 5 K 6 γ γ 2 p 1 1 p 1 1 p 1 1 γ γ 4 a 1 1 a p 1 γ 5 q 1 q γ 6 a 2 1 q 1 p Hee, a 1 = (q 1)( 1). Let F q, = a, b : a q = b = 1, b 1 ab = a u. Conside L 1 = {id Zp } and L 2 = Z p \ L 1 ae the ational conjugacy classes of Z p, and T 1 = {id Fq, }, T 2 = {a,..., a q 1 } and, T 3 = F q, \ a ae the ational conjugacy classes of F q,. Then K 1 = L 1 T 1, K 2 = L 2 T 1, K 3 = L 1 T 2, K 4 = L 2 T 2, K 5 = L 1 T 3, and K 6 = L 2 T 3. Table 12. Rational Chaacte Table of Goup G 5. QCT (G 5 ) K 1 K 2 K 3 K 4 K 5 γ γ γ 3 q 1 q 1 q γ 4 a 1 a 1 1 q 1 1 γ 5 p Hee, a 1 = (p 1)( 1) and a 2 = (p 1)(q 1). Let F p,q = a, b : a p = b q = 1, b 1 ab = a u. Then, the ational conjugacy classes ae, K 1 = id G5, K 2 = {a i : 1 i p 1}, K 3 = 1 i=1 (biq ) G 5, K 4 = q 1 i=1 (bi ) G 5 and K 5 = (k,q)=1 (b k ) G 5.

18 168 SHABANI, ASHRAFI, AND GHORBANI Table 13. Rational Chaacte Table of Goup G = G i+5, 1 i 1. QCT (G i+5 ) K 1 K 2 K 3 K 4 K 5 γ γ γ 3 q 1 q γ 4 p 1 1 p γ 5 (p 1)(q 1) q + 1 p Hee, K 1 = {id Gi+5 }, K 2 = l 1 i=1 (a v i ) G, K 3 = l 2 i=1 (b u i ) G, K 4 = l 3 i=1 (b u i a v i ) G and K 5 = 1 k=1 (ck ) G ae the ational conjugacy classes of G and γ 1 = 1, γ 2 = 1 n=1 χ n, γ 3 = l 3 s=1 η s, γ 4 = l 1 l=1 θ l and γ 5 = l 2 m=1 φ m. Acknowledgments The authos ae indebted to the anonymous efeees fo thei comments and helpful emaks. We ae also vey gateful to pofesso Thomas Beue fo giving us a poof fo Lemma 1.1. The fist and second authos wee patially suppoted by the Univesity of Kashan unde gant numbe /5, and the thid autho was patially suppoted by the Shahid Rajaee Teache Taining Univesity unde gant numbe Refeences 1. H. Behavesh, Quasi-pemutation epesentations of F p,q, Vietnam J. Math. 31 (2003), J. H. Conway, R. T. Cutis, S. P. Noton, R. A. Pake, R. A. Wilson, Atlas of Finite Goups, Oxfod Univ. Pess, Claendon, Oxfod, H. Hölde, Die Guppen de Odnungen p 3, pq 2, pq, p 4, Math. Ann. XLIII (1893), I. M. Iscaacs, Chaacte Theoy of Finite Goups, Dove, New-Yok, G. James, M. Liebeck, Repesentations and Chaactes of Goups, Cambidge Univ. Pess, London-New Yok, The Gap Team, GAP Goups, Algoithms and Pogamming, Vesion T. Yamada, On the goup algebas of metabelian goups ove algebaic numbe fields I, Osaka J. Math. 6 (1969), H. Shabani Depatment of Pue Mathematics, Faculty of Mathematical Sciences, Univesity of Kashan, Kashan , I. R. Ian.

19 RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS A. R. Ashafi Depatment of Pue Mathematics, Faculty of Mathematical Sciences, Univesity of Kashan, Kashan , I. R. Ian. M. Ghobani Depatment of Mathematics, Faculty of Science, Shahid Rajaee Teache Taining Univesity, Tehan, , I. R. Ian.

20 Jounal of Algebaic Systems RATIONAL CHARACTER TABLE OF SOME FINITE GROUPS H. Shabani, A. R. Ashafi and M. Ghobani حسین شبانی ١ سید علیرضا اشرفی ١ و مجتبی قربانی ٢ ١ کاشان دانشگاه کاشان دانشکده علوم ریاضی گروه ریاضی محض کد پستی ٨٧٣١٧۵٣١۵٣ ٢ تهران دانشگاه تربیت دبیر شهید رجایی دانشکده علوم پایه گروه ریاضی کد پستی - ١۶٧٨۵ ١٣۶ در این مقاله به محاسبه جدول سرشتهای گروه دودوری T ۴n گروههای مرتبه pq و گروههای مرتبه pq پرداختهایم. همچنین برخی از خواص جدول گویا-سرشت نیز آورده شدهاست. کلمات کلیدی: جدول سرشتهای گویا جدول سرشت گروه گالوا. ۶