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A220470 Smallest order of a group with an irreducible complex representation of dimension n.

%I A220470 #125 Apr 09 2020 17:07:33
%S A220470 1,6,12,20,55,42,56,72,144,110,253,156,351,336,240,272,1751,342,3420,
%T A220470 500,672,506,1081,600,2525,702,1512,812,1711,930,992,1440
%N A220470 Smallest order of a group with an irreducible complex representation of dimension n.
%C A220470 a(1) = 1 because the trivial group works.
%C A220470 If n > 1, then let G_n be a group of minimal order which has an irreducible complex representation of dimension n.
%C A220470 Since G_n has the n-dimensional and trivial representations, a_n = |G_n| >= n^2 + 1.
%C A220470 Since n divides the order of any finite group with an irreducible complex representation of dimension n, n divides a_n and this allows us to strengthen the lower bound to a_n >= n^2 + n.
%C A220470 An upper bound is given by a_n <= n*q_n, where q_n is the smallest positive integer which is a power of a prime and which is congruent to 1 mod n. This is because the group of affine transformations x --> a*x + b (from the finite field F_(q_n) to itself), where a^n = 1 and b is an arbitrary element of F_(q_n), has order n*q_n and an irreducible complex representation of dimension n.
%C A220470 The upper bound and the lower bound coincide if and only if (n > 1 and) n+1 is a power of a prime. This means that, for any such n, a(n) = n^2 + n.
%C A220470 Upper bound is n*A224503(n), for n > 1.
%C A220470 This function is sub-multiplicative: a(m*n) <= a(m)*a(n) because (any) G_m x G_n has an irreducible complex representation of dimension m*n.
%D A220470 Marshall Hall, Jr., Theory of Groups, AMS Chelsea Publishing, 1999 reprinting, pp. 136-138.
%D A220470 I. Martin Isaacs, Finite Group Theory, AMS, 2008, p. 35.
%D A220470 Gordon James and Martin Liebeck, Representations and Characters of Groups, Cambridge University Press, 1995 reprinting, pp. 217-218.
%H A220470 Sean Cleary, Roland Maio, <a href="https://arxiv.org/abs/2001.06407">Counting difficult tree pairs with respect to the rotation distance problem</a>, arXiv:2001.06407 [cs.DS], 2020.
%H A220470 David L. Harden, <a href="/A220470/a220470.pdf">The smallest group with an irreducible representation of dimension 17</a>
%H A220470 David L. Harden, <a href="/A220470/a220470_1.pdf">The smallest group with an irreducible representation of dimension 19</a>
%H A220470 David L. Harden, <a href="/A220470/a220470_2.pdf">When is the lower bound sharp?</a>
%e A220470 For n=5, the lower bound is a(5) >= 30, while the upper bound is a(5) <= 55.
%e A220470 So one only needs to prove that no group of order 30, 35, 40, 45, or 50 has an irreducible complex representation of dimension 5.
%e A220470 All groups of order 35 and 45 are Abelian, so all their irreducible complex representations have dimension 1.
%e A220470 For orders 30, 40, and 50, recall that the dimension of an irreducible complex representation is bounded from above by the index of an Abelian subgroup.
%e A220470 A group of order 50 has an Abelian subgroup of index 2.
%e A220470 A group of order 40 has a normal Sylow 5-subgroup P, which must be centralized by an element t of order 2 in a Sylow 2-subgroup. Then <P,t> is a cyclic subgroup of index 4.
%e A220470 A group of order 30 has a cyclic subgroup of index 2.
%e A220470 All these indices are too low to allow such a group to have an irreducible complex representation of dimension 5, and the a(5) = 55 result is proved.
%e A220470 Comments from Gabriele Nebe (Lehrstuhl D für Mathematik, RWTH Aachen), added by _N. J. A. Sloane_, Apr 13 2013; extended by _David L. Harden_, Nov 08 2017: (Start)
%e A220470 The following are examples of groups that realize the given values of a(n).
%e A220470 1=C_1, 6=S_3, 12=A_4, 20=C_5:C_4, 55=C_11:C_5, 42=C_7:C_6, 56=(C_2xC_2xC_2):C_7, 72=C_3xC_3:(some regular subgroup of L_2(3) on 8 points), 144=A_4xA_4, 110 = C_11:C_10, 253=C_23:C_11, 156=C_13:C_12, 351=(C_3xC_3xC_3):C_13, 336 = (C_2xC_2xC_2):C_7 x S_3, 240 = (C_2xC_2xC_2xC_2):C_15, 272 = C_17:C_16, 1751 = C_103:C_17, 342 = C_19:C_18, 3420 = PSL_2(19).
%e A220470 If one has a small cyclic group of order p where phi(p) is a multiple of n, then take C_p:C_n. For other n it is reasonable to take elementary abelian groups of order n+1 that admit a subgroup of their automorphism group acting regularly on the n nontrivial characters. (End)
%Y A220470 Cf. A224503.
%K A220470 nonn,more
%O A220470 1,2
%A A220470 _David L. Harden_, Apr 10 2013
%E A220470 a(17)-a(18) from _David L. Harden_, Nov 08 2017
%E A220470 a(19) from _David L. Harden_, Dec 08 2017
%E A220470 a(20)-a(31) from _Bob Heffernan_, Jul 06 2018
%E A220470 a(32) from _Bob Heffernan_, Nov 15 2019