Proc. Indian Acad. Sci. (Math. Sci.) Vol. 119, No. 2, April 2009, pp. 137–143. © Printed in India
A note on p-groups of order ≤ p4
HOUSHANG BEHRAVESH1 and HAMID MOUSAVI2
1Faculty of Science, Department of Mathematics, University of Urmia, Urimia, Iran 2Department of Mathematics, University of Tabriz, Tabriz, Iran E-mail: [email protected]; [email protected]
MS received 21 January 2008; revised 16 February 2008
Abstract. In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G|≤p4, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.
Keywords. Quasi-permutation representations; p-groups; character theory.
1. Introduction By a quasi-permutation matrix, we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi- permutation matrices over the rational field Q, and c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices (see [1]). By a rational valued character we mean a character χ corresponding to a complex rep- resentation of G such that χ(g) ∈ Q for all g ∈ G. As the values of the character of a complex representation are algebraic integers, a rational valued character is in fact integer valued. A quasi-permutation representation of G is then simply a complex representation of G whose character values are rational and non-negative. The module of such a represen- tation will be called a quasi-permutation module. We will call a homomorphism from G to GL(n, Q) a rational representation of G and its corresponding character will be called a rational character of G. It is easy to see that
c(G) ≤ q(G) ≤ p(G), where G is a finite group. Now we would like to state a problem from Prof. Brian Hartley (1992–94).
Problem. Let G be a finite p-group. Find G such that
c(G) = q(G) = p(G).
In [4] we showed that, when G is a generalized quaternion group then
2c(G) = q(G) = p(G).
137 138 Houshang Behravesh and Hamid Mousavi
So in this case a good question to be asked is: For a 2-group G, when is c(G) < q(G) < p(G)? When G is a finite 2-group of order ≤ 64 we calculate c(G), q(G) and p(G) (see [6] and [7]). From now we will assume that p is an odd prime. In fact it is easy to prove that c(G) = q(G). So in this case a good question to be asked is: For a p-group G with p an odd prime, when is q(G) = p(G)? In [2], [3] and [5] we have calculated c(G), q(G) and p(G) for an abelian and the metacyclic p-groups and we have showed that c(G) = q(G) = p(G). In [1] we have showed that for groups of class 2 and cyclic center, c(G) = q(G) = p(G). So for non- abelian groups of order p3, again c(G) = q(G) = p(G). Since the classification of groups of order p4 depends on p to be 3 or greater than 3 (see [8]), for the case p = 3, we use direct calculation by using GAP [11], for character tables and subgroups and also the algorithms from [1]. Hence we will have the following table:
G c(G) = q(G) = p(G) (81, 3) 18 (81, 4) 18 (81, 6) 27 (81, 7) 9 (81, 8) 27 (81, 9) 27 (81, 10) 27 (81, 12) 12 (81, 13) 12 (81, 14) 27
In this paper we will show that for a finite non-metacyclic p-group G of order p4, c(G) = q(G) = p(G). Finally this shows that for any p-group, G with |G|≤p4, c(G) = q(G) = p(G). By Corollary 4.4 of [1], we know that for any finite p-groups G, c(G) = q(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open. We will need the following notation and definition in order to calculate c(G). Notation. Let (χ) be the Galois group of Q(χ) over Q. DEFINITION 1.1 Let G be a finite group. Let χ be an irreducible complex character of G. Then define =| | (1) d(χ) (χ) χ(1). 0, if χ = 1G, (2) m(χ) = | { χ α(g) g ∈ G}|, . min α∈(χ) : otherwise = α + (3) c(χ) α∈(χ) χ m(χ)1G. A note on p-groups of order ≤ p4 139
2. A note on groups of order p4 Let p denote an odd prime. In this section we will give some properties of non-abelian groups of order p4. Lemma 2.1. Let G be a finite non-abelian p-group of order p4. Then (a) |Z(G)|=p or p2; (b) cd(G) ={1,p}, where cd(G) denote the set of irreducible character degrees; (c) |G|≤p2.
Proof. (a) It is known that |G: Z(G)| is divisible by p2 and Z(G) = 1. So the result follows. (b) By (a), |G: Z(G)|≤p3. So by Corollary 2.30 and Theorem 3.12 of [9], the result follows. (c) Let |G|=p3. Then by Theorem 2.13 of [9], we have p4 = p + p2α. This equation has no integer solution. So the result follows. 2
Lemma 2.2. Let G be a finite non-abelian p-group of order p4.IfZ(G) is cyclic of order p, then G has order p2. Moreover Z(G) 3. Groups with elementary abelian center and (G) = Z(G) Let |G|=p4 be a non-abelian group. It is easy to see that exp(G) = p, p2 or p3.IfG has an element of order p3, then G is metacyclic. Hence c(G) = q(G) = p(G) (see [3] and [5]). So we will assume that exp(G) = p or p2. ∼ Let Z(G) = Cp ×Cp. Since G/Z(G) is an elementary abelian group, so (G) ≤ Z(G), where (G) denote the Frattini subgroup of G. Let (G) = Z(G). Then d(G) = 2, where d(G) denote the minimal numbers of generators of the group G. Now let G =x,y. Since G ≤ Z(G),soG =[x,y]. But Z(G) is an elementary abelian group. So |G|=p. Let o(x) = o(y) = p2. = ∼ (a) If x y 1, then G = Cp2 Cp2 , that is, 2 2 − + G =x,y: xp = yp = 1,y 1xy = x1 p. AgainG is metacyclic and we know that c(G) = q(G) = p(G). (b) If x y = 1, then xp = yp. Therefore (xy−1)p = xpy−p[y−1,x]p(p+1)/2. Since −1 p −1 p is odd, so (xy ) = 1. Without loss of generality we can change y by xy and let o(x) = p2 and o(y) = p. Since |G|=|Gp|=p,soG Gp = 1. Hence 2 G =x,y,z: xp = yp = zp = [x,z] = [y,z] = 1, [x,y] = z. By an easy calculation one can see that G =z and Z(G) =xpz. Hence = ∼ × G/G xG yG = Cp2 Cp. It is easy to see that, for any linear character χ of G, d(χ) is equal to p(p − 1) or p − 1 and for non-linear irreducible characters of G say χ, d(χ) = p(p − 1). By Theorem 2.4, we are able to add only two non-linear irreducible characters and get a faithful character. The characters of this type are similar to characters of abelian group Z(G) with an exceptional case that xp is not the kernel of any non-linear irre- ducible character of G. But it is easy to see that like abelian groups of type Cp × Cp, c(G) = q(G) ≤ 2p2. The other way of finding a faithful character is to use one from non-linear irreducible characters and at least one from linear characters. We know that all subgroups of order p2 of G/G are the following groups: (1) xpGyG; (2) xyαG, where 0 ≤ α ≤ p − 1. A note on p-groups of order ≤ p4 141 Also all subgroups of order p of G/G are the following groups: (1) yG; (2) yxαp G, where 0 ≤ β ≤ p − 1. It is easy to see that xpG is a subgroup of any subgroups of order p2 of G/G. Hence to find a faithful character, by using a non-linear irreducible say χ, and a linear character say η, we have to use only a linear character of G/G with kernel of order p. In this case 2 d(χ)+d(η) = p(p−1)+p(p−1). Hence d(χ)+d(η) = 2p −2p. By using Lemma 4.5 α + β of [1], it is easy to see that, the minimal value for α∈(χ) χ β∈(η) η is equal to 2 −2p. So Theorem 3.6 and Corollary 3.14 of [1] shows that c(G) = q(G) = 2p . Let H =x and K =y,z. It is easy to see that HG KG = 1. So by Lemma 2.1 of [1], p(G) ≤ 2p2. Hence c(G) = q(G) = p(G). Therefore we have the following theorem. 4 ∼ Theorem 3.1. Let G be finite p-group of order p . Also let Z(G) = Cp × Cp and let (G) = Z(G). Then c(G) = q(G) = p(G). 4. Groups with elementary abelian center and (G) < Z(G) Let |G|=p4 be a non-abelian group and Z(G) be an elementary abelian group. Also let (G) < Z(G). Then d(G) = 3. Let G =x,y,z. Since |G: Z(G)|=p2, we can assume that z ∈ Z(G). Therefore |Gp|=1orp. In both cases as in case (b) of the previous section we can assume that, z, y are of order p. Hence G is the direct product of a group of order p by a non-abelian group of order p3. Hence we have the following groups: p p p G1 =x,y,t: x = y = t = [x,t] = [y,t] = 1, [x,y] = tz or p2 p p G2 =x,y: x = y = 1, [x,y] = x z. Therefore Z(G1) =tz , p Z(G2) =x z G =t 1 and = p G2 x = =∼ × × = Gi/Gi xGi,yGi,zGi Cp Cp Cp (i 1, 2). = = = = Let A y z , B x,t z and C x z . Then AG1 BG1 1 and AG2 = ≤ 2 + = CG2 1. Hence by Lemma 2.1 of [1], p(Gi) p p, for i 1, 2. = = 2+ Let G G1. We will show that c(G) p p. As in the previous case we are able to use only two non-linear irreducible characters say χ1 and χ2, such that, ker χ1 ker χ1 = 1. + = 2 − But in this case we will have d(χ1) d(χ2) 2p 2p, and the minimal value for χ α + χ β is equal to −2p. Since c(G) ≤ p(G) ≤ p2 + p,wehaveto α∈(χ1) 1 β∈(χ2) 2 use one linear and one non-linear irreducible character to find a faithful character. 142 Houshang Behravesh and Hamid Mousavi We know that t is not the kernel of any non-linear irreducible character of G. Let χ be a non-linear irreducible character of G, with kernel equal to z. Let η be a linear character of G/G with kernel equal to xG ,yG. Then it is easy to see that ker χ ker η = 1. + = − + − α + β Also d(χ) d(η) p(p 1) (p 1). The minimal value for α∈(χ) χ β∈(η) η is equal to −p − 1. So c(G) = p2 + p. 2 By a similar calculation one can prove that when G = G2, then c(G) = p + p.Sowe have the following theorem. 4 ∼ Theorem 4.1. Let G be finite p-group of order p . Also let Z(G) = Cp × Cp and let (G) < Z(G). Then c(G) = q(G) = p(G). 5. Groups with cyclic center of order p>3 In this section we will assume that G is a p-group of order p4, where p>3 and |Z(G)|=p. In this case we have the following groups of order p4 (see [8]): 2 − + − − (1) x,y,z: xp = yp = zp = 1,y 1xy = x1 p,z 1xz = xy, z 1yz = xy; (2) x,y,z,t: xp = yp = zp = tp = 1, [x,y] = [x,z] = [t,x] = [z, y] = 1, t−1zt = zy, t −1yt = yx; 2 − + − − (3) x,y,z: xp = yp = zp = 1,y 1xy = x1 p,z 1xz = xy, z 1yz = xpy; 2 − + − − (4) x,y,z: xp = yp = zp = 1,y 1xy = x1 p,z 1xz = xy, z 1yz = xαp y, where α = any quadratic non-residue modp. Since Z(G) is cyclic, we have to find a core-free subgroup with small index by Corol- lary 2.4 of [1]. Also note that we have a core-free cyclic subgoup of order p and groups of order p3 are normal in G.Sop(G) = p2 or p3. It is easy to see that H =y,z is a core-free subgroup of types (1) and (2). So p(G) = p2 for the groups of types (1) and (2). Also z is a core-free sungroup of types (3) and (4). So p(G) ≤ p3 for the groups of types (3) and (4). Let p(G) = p2. Since Z(G) is cyclic, so by Theorem 2.32(b) of [9], G has a faith- ful character χ of degree p. Also by Lemma 2.27(c) of [9], χz = pω, where ω is a 2 p-th primitive root of unity. For such a character d(χ) ≥ p(p − 1) = p − p. Also ≥− ∈ −{ } ≥ 2 α∈(G) χ(g) p, where g Z(G) 1 .Soc(χ) p . Now by Corollary 3.11 of [1], it follows that c(G) ≥ c(χ) ≥ p2. Since p2 ≤ c(G) ≤ q(G) ≤ p(G) ≤ p2. Now let G be isomorphic to type (1). One can prove = ∼ × H zx, y = Cp2 Cp. Since G is a p-group, hence it is an M-group. So any character of G is monomial. Let us induce a character of H, say χ = χ11y, A note on p-groups of order ≤ p4 143 G where χ1 is a faithful character of zx. Then it is easy to see that χ is a faithful character of G. Also Z(G) G, so we must have a non-linear character of degree p, which is not faithful. In this case we will have at least p(p − 1) faithful irreducible characters in one Galois orbit. Also note that by Lemma 2.27(c) of [9], χz = pω, where ω is a p-th primitive root of unity. Hence c(G) = c(χ) + m(χ). But c(χ) = p(p(p − 1)) = p3 − p2 and by considering an element of Z(G), one can see that m(χ) ≥ p2.Soc(G) ≥ p3. Hence c(G) = q(G) = p(G) = p3. Finally we are able to produce a similar proof for type (4). Therefore we have the following theorem: Theorem 5.1. Let G be a group of order p4 and |Z(G)|=p. Then c(G) = q(G) = p(G). Acknowledgement The authors are grateful to the referee for his valuable suggestions and comments. The paper was revised according to his suggestions. The authors also wish to thank the Urmia University Research Council for financial support. References [1] Behravesh H, Quasi-permutation representations of p-groups of class 2, J. London Math. Soc. 55(2) (1997) 251–260 [2] Behravesh H, The minimal degree of a faithful quasi-permutation representation of an abelian group, Galsgow Maths. J. 39 (1997) 51–57 [3] Behravesh H, Quasi-permutation representations of metacyclic 2-groups, J. Sci. I. R. Iran. 9(3) (1998) 258–264 [4] Behravesh H, Quasi-permutation representations of metacyclic 2-groups with cyclic cen- ter, Bull. Iranian Math. Soc. 24(1) (1998) 1–14 [5] Behravesh H, Quasi-permutation representations of metacyclic 2-groups with non-cyclic center, Southeast Asian Bull. Math. Springer-Verlag 24 (2000) 345–353 [6] Behravesh H and Ghaffarzadeh G, Quasi-permutation representations of 2-groups of order ≤ 32, Far East J. Math. Sci. (FJMS) 23(3) (2006) 361–367 [7] Behravesh H and Ghaffarzadeh G, Quasi-permutation represntations of groups of order 64, Turk J. Math. Suppl. 31 (2007) 1–6 [8] Burnside W, Theory of groups of finite order (Cambridge University Press) (1897) [9] Isaacs I M, Character theory of finite groups (New York: Academic Press) (1976) [10] RobinsonDJS,Acourse in the theory of groups (New York: Springer) (1982) [11] Schonert M et al, GAP: Groups, Algorithms and Programming, Lehrstuhl D fur¨ Mathematik (RWTH Aachen) (1994)