MINIMAL GROUP DETERMINANTS FOR DICYCLIC GROUPS BISHNU PAUDEL AND CHRISTOPHER PINNER Abstract. We determine the minimal non-trivial integer group determinant for the dicyclic group of order 4n when n is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order 4p. 1. Introduction For a finite group G = {g1,...,gn} of order n we assign a variable xg for each D element g ∈ G and define the group determinant G(xg1 ,...,xgn ) to be the deter- minant of the n × n matrix whose ijth entry is x −1 . We are interested here in gigj the values that the group determinant can take when the variables are all integers D S(G)= { G(xg1 ,...,xgn ): xg1 ,...,xgn ∈ Z}. Notice that S(G) will be closed under multiplication: D D D (1) G(ag1 ,...,agn ) G(bg1 ,...,bgn )= G(cg1 ,...,cgn ), cg = aubv. uv=g X An old problem of Olga Taussky-Todd is to determine S(Zn), where the group determinants are the n × n circulant determinants with integer entries. Here and throughout we write Zn for the integers modulo n, and p will always denote a prime. Laquer [11] and Newman [16, 17] obtained divisibility conditions on the values of the group determinant for integer variables for cyclic groups and a complete description of the values for certain cyclic groups. For example, Laquer [11] and Newman [16] showed that a (2) S(Zp)= {p m, (m,p)=1, a =0 or a ≥ 2}, and Laquer [11] that for odd p arXiv:2102.04536v1 [math.NT] 8 Feb 2021 a b (3) S(Z2p)= {2 p m, (m, 2p)=1, a =0 or a ≥ 2, b =0 or b ≥ 2}. Newman [17] described S(Z9) with upper and lower set inclusions for general Zp2 . For the general cyclic group Newman [16] showed that (4) {m ∈ Z : gcd(m,n)=1} ⊂ S(Zn), with a divisibility restriction for the values not coprime to the order: t t+1 (5) p k n, p | m ∈ S(Zn) ⇒ p | m. Date: February 10, 2021. 2010 Mathematics Subject Classification. Primary: 11R06, 15B36; Secondary: 11B83, 11C08, 11C20, 11G50, 11R09, 11T22, 43A40. Key words and phrases. group determinant, dicyclic group, Lind-Lehmer constant, Mahler measure. 1 2 B. PAUDEL AND C. PINNER For odd p the values for the Dihedral groups of order 2p or 4p were obtained in [2]: a b S(D2p)= {2 p m : (m, 2p)=1, a =0 or a ≥ 2, b =0 or b ≥ 3}, 3 S(D4p)= {m ≡ 1 mod 4 : p ∤ m or p | m} ∪{2apbm : (m, 2p)=1, a =4 or a ≥ 6, b =0 or b ≥ 3}, with a counterpart to (4) (6) {m ∈ Z : gcd(m, 2n)=1} ⊂ S(D2n), for n odd, but only those gcd(m, 2n) = 1 with m ≡ 1 mod 4 when n is even, and the divisibility condition (5) t 2t+1 (7) p k n, p | m ∈ S(D2n) ⇒ p | m, for odd p, with 22, 24 or 22t+4 | m when p = 2 and t =0, 1 or t ≥ 2 respectively. A complete description for all groups of order at most 14 was found in [19] and for S4 in [21]. For example for the two dicyclic groups of order less than 14: 2 8 (8) S(Q8)= {8m +1, (8m − 3)p , and 2 m : m ∈ Z, p ≡ 3 mod 4} and a b (9) S(Q12)= {2 3 m : a =0, 4 or a ≥ 6,b =0 or b ≥ 3, gcd(m, 6)=1} ∪{253bm : b =4 or b ≥ 6, gcd(m, 6)=1} ∪{253bmp : b =0, 3 or5, gcd(m, 6)=1,p ≡ 5 mod 12} ∪{253bmp2 : b =0, 3 or 5, gcd(m, 6)=1, p ≡ 5 mod 6}. The complexity encountered even for small groups [19] makes it clear that obtaining S(G) is not in general feasible. Indeed simply finding the smallest non-trivial integer determinant D (10) λ(G) := min{| G(xg1 ,...,xgn )|≥ 2 : xgi ∈ Z} can be difficult. For a group of order n taking xe = 0 and xg = 1 for g 6= e always gives determinant (−1)n−1(n − 1), so we have as our trivial bound (11) λ(G) ≤ |G|− 1 for |G|≥ 3, with λ({e}) = 2, λ(Z2) = 3. 3 Kaiblinger [9] obtained λ(Zn) when 420 ∤ n, with this extended to 2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 ∤ n in [18]. Values of λ(G) for non-cyclic abelian G are considered in [6, 7, 20, 3, 15]. In [2] the value of λ(D2n) was obtained for any dihedral group of order 2n with 22 · 32 · 5 · 7 · 11 · 13 · ... · 107 · 109 · 113 ∤ n. Our goal here is to determine similar results for Q4n, the dicyclic group of order 4n, when n is odd. 2. Lind Mahler Measure For a polynomial F ∈ Z[x, x−1] one defines the traditional logarithmic Mahler measure by 1 (12) m(F )= log |F (e2πiθ)|dθ. Z0 MINIMAL GROUP DETERMINANTS FOR DICYCLIC GROUPS 3 Lind [14] regarded this as a measure on the group R/Z and extended the concept to a compact abelian group with a Haar measure. For example for an F ∈ Z[x, x−1] and cyclic group Zn we can define a Zn-logarithmic measure 1 mZ (F )= log |F (z)| n n zn=1 X 1 Z Z That is m n (F )= n log |M n (F )| where n−1 j 2πi/n MZn (F ) := F (wn), wn := e . j=0 Y More generally for a finite abelian group (13) G = Zn1 ×···× Znk we can define a logarithmic G-measure on Z[x1,...,xk] by n −1 n −1 1 1 k m (F )= log |M (F )|, M (F )= ··· F wj1 , ··· , wjk . G |G| G G n1 nk j1=0 j =0 Y Yk As by observed by Dedekind the group determinant for a finite abelian group can be factored into linear factors using the group characters Gˆ D (14) G(xg1 ,...,xgn )= (χ(g1)xg1 + ··· + χ(gn)xgn ) , χY∈Gˆ and can be related directly to a Lind Mahler measure for the group, see for example [20]. For example in the cyclic case, see [10] D n−1 Zn (a0,a1,...,an−1)= MZn (a0 + a1x + ··· + an−1x ), and in the general finite abelian case (13) D (a ,...,a )= M a xj1 ··· xjk . G g1 gn G g 1 k g=(j1X,...,jk)∈G For a finite non-abelian group the group determinant will not factor into linear factors but can still be factored using the group representations Gˆ deg(ρ) D (x ,...,x )= det x ρ(g) G g1 gn g ρY∈Gˆ gX∈G as discovered by Frobenius, see for example [8, 4]. In [2] it was shown that the group determinants for the dihedral group of order 2n, n 2 −1 n−1 n−1 D2n = hx, y : x =1,y =1, xy = yx i = {1, x, ··· , x ,y,yx,...,yx }, can be written as a Zn-measure D −1 −1 (15) D2n (a0,...,an−1,b0,...,bn−1)= MZn f(x)f(x ) − g(x)g(x ) , where n−1 n−1 (16) f(x)= a0 + ··· + an−1x , g(x)= b0 + ··· + bn−1x . Similarly for the dicyclic group of order 4n, 2n 2 n −1 2n−1 2n−1 Q4n = hx, y : x =1,y = x , xy = yx i = {1, x, ··· , x ,y,yx,...,yx }, 4 B. PAUDEL AND C. PINNER it was shown in [19] that the group representations give D −1 n −1 (17) Q4n (a0,...,a2n−1,b0,...,b2n−1)= MZ2n f(x)f(x ) − x g(x)g(x ) , where 2n−1 2n−1 (18) f(x)= a0 + ··· + a2n−1x , g(x)= b0 + ··· + b2n−1x . Notice that we can conversely use the group determinant to define a Lind style polynomial measure for non-abelian finite groups. For example we can define D2n and Q4n measures on Z[x, y] by −1 −1 MD2n (f(x)+ yg(x)) = MZn f(x)f(x ) − g(x)g(x ) , −1 n −1 Z (19) MQ4n (f(x)+ yg(x)) = M 2n f(x)f(x ) − x g(x)g(x ) , although here the polynomial ring is no longer commutative, the monomials having to satisfy the group relations y2 = 1, xy = yx−1 etc., the relations allowing us to reduce any F (x, y) to the form f(x)+ yg(x), with f and g of the form (16) or (18) if we want, and to multiply and reduce two polynomials. The classical Lehmer problem [12] is to determine inf{m(F ) > 0 : F ∈ Z[x]}. Given the correspondence between the Lind measures and group determinants in the abelian case we can regard determining λ(G) for a finite group as the Lind-Lehmer problem for that group. An alternative way of extending the Mahler measure to groups can be found in [5]. 3. Minimal determinants for odd n For the dicyclic groups G = Q4n we have some extra properties when n is odd. For example, since n (20) MG(f(x)+ yg(x)) = (−1) MG(g(x)+ yf(x)), if n is odd we have −m ∈ S(G) whenever m ∈ S(G). This is certainly not true when n is even as we saw for Q8. When n is odd we also have n (n−1)/2 n (n−1)/2 MG 1 + (x + 1)(x + ··· + x )+ y(x + 1)(x + ··· + x ) =2n − 1, always improving on the trivial bound (11), and 2 (21) MG(x +1)= 16, giving us an absolute bound λ(Q4n) ≤ 16 for n odd.