Dear Rulin, Max, everybody,
Rulin Shen's question is exactly equivalent to asking which groups
contain a "Hadamard difference set": this is a set S of size 2n^2 - n in
a group of order 4n^2 such that each non-identity element of the group
has n^2 - n representations as ab^-1, for a, b in S.
(I can provide a proof of this if anyone is interested.)
There is quite a lot of literature on Hadamard difference sets. Here are
a few relevant papers; there are many more!
David Gluck, Hadamard difference sets in groups of order 64, J.
Combinatorial
Theory (A) 51 (1989), 138-140
Alec Biehl et al., Finding Hadamard difference sets,
http://www.sci.sdsu.edu/math-reu/2013-1.pdf
Omar Abughneim, On (64,28,12) difference sets, Ars Combinatoria 111 (2013),
401-419.
Dylan Peifer, Hadamard difference sets,
http://www.math.cornell.edu/~djp282/documents/olivetti-2016.pdf
Dieter Jungnickel and Bernhard Schmidt, Difference sets: an update,
http://www.ntu.edu.sg/home/bernhard/Publications/pub/update1.pdf
Peter.
On 02/02/17 16:31, Max Horn wrote:
Dear Rulin,
On 02 Feb 2017, at 12:45, Rulin Shen <shenru...@hotmail.com> wrote:
Dear Prof. Cameron,
Thanks for your answer. Sorry to my question's condition should be |H^c \cap
Hg|=|G|/4, where H^c the complement of H in G, and all g. So sorry!
Then the problem seems to become trivial: H^c \cap Hg either equals the empty
set (when g\in H) or else Hg. By your condition, the former case must not occur
for any g\neq 1, thus H must be trivial. But then |H^c \cap Hg|=1 for g\neq 1,
thus |G|=4 and G is either C_4 or C_2 x C_2.
Cheers,
Max
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