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)<G and G/Z(G) is not abelian. Proof. Let G has order p. Then it is obvious that G = Z(G). Since Z(G) is cyclic and Z(G) = G, so by Lemma 4.3 of [1], |G: Z(G)|=p2n for some integer n. But this is not possible, so the result follows. 2 ≤ = −1 Note. Let H G. Then define HG x∈G x Hx. It is called the core of H in G. Theorem 2.3. Let G be a finite non-abelian p-group of order p4. Let G have cyclic center of order p2. Then c(G) = q(G) = p(G) = p3. Proof. Since Z(G) is cyclic and G is not abelian so, by 5.3.6 of [10], there exists an element of order p, say x ∈ G such that x ∈ Z(G) and xG = 1. So by Corollary 2.4 of [1], p(G) ≤ p3. Since Z(G) is cyclic and |G: Z(G)|=p2, so by Lemma 3.8 of [1], we have a faithful irreducible character χ, and by Corollary 2.30 of [9], χ vanishes on G − Z(G).Nowby Lemma 2.27 of [9], and Corollary 3.11 of [1], it is easy to see that p3 = c(G) ≤ q(G) ≤ p(G) ≤ p3. So the result follows. 2 Theorem 2.4. LetG be a finite group and G < Z(G) and Z(G) = AG for some A ≤ Z(G) and A G = 1. Also let cd(G) ={1, |G: Z(G)|1/2}. Then we have |A|(|G |−1), non-linear characters. Let ηj denote the linear characters of A, for 1 ≤ j ≤|A| and ψk denote the non-trivial characters of G, for 2 ≤ k ≤|G|. Then 1/2 |G: Z(G)| ηj (x)ψk(y), g = xy ∈ Z(G) x ∈ A, y ∈ G , χj,k(g) = 0,g∈ G − Z(G), denote all the non-linear characters of G. 140 Houshang Behravesh and Hamid Mousavi Proof. Let G < Z(G) and also we let Z(G) = AG for some A ≤ Z(G) and A G = 1. Let cd(G) ={1, |G: Z(G)|1/2}. Then it is easy to see that the number of non-linear characters of G are equal to |A|(|G|−1). Also the characters of Z(G) can be calculated by the characters of A and G. Since the trivial character of Z(G) multiplied by |G: Z(G)|1/2 is not a character of G, so it is easy to see that the non-linear characters of G can be calculated from the characters of A times by the non-trivial characters of G and times by |G: Z(G)|1/2. So the result follows. 2 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.