View source for Abelian difference set

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100201.png" /> be a [[Group|group]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100203.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100204.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100205.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100207.png" />-difference set of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100208.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a1100209.png" /> if every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002011.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002012.png" /> different representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002014.png" />, see [[#References|[a1]]]. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002015.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002016.png" />-difference set in the [[Cyclic group|cyclic group]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002018.png" /> is Abelian (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002021.png" /> are equivalent if there is a group [[Automorphism|automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002023.png" />. The existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002024.png" />-difference set is equivalent to the existence of a symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002025.png" />-design with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002026.png" /> acting as a regular automorphism group (cf. also [[Difference set|Difference set]]). If two difference sets correspond to isomorphic designs, the difference sets are called isomorphic. Given certain parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002029.png" /> and a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002030.png" />, the problem is to construct a difference set with those parameters or prove non-existence. To prove non-existence of Abelian difference sets, results from [[Algebraic number theory|algebraic number theory]] are required: The existence of the difference set implies the existence of an [[algebraic integer]] of absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002031.png" /> in some [[Cyclotomic field|cyclotomic field]]. In several cases one can prove that no such element exists, see [[#References|[a5]]]. Another approach for non-existence results uses multipliers: A multiplier of an Abelian difference set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002032.png" /> is an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002035.png" />. A statement that certain group automorphisms have to be multipliers of putative difference sets is called a multiplier theorem. It is known, for instance, that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002036.png" /> is a multiplier of an Abelian difference set provided that: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002037.png" /> divides the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002038.png" />; ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002039.png" /> is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002040.png" />; and iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002041.png" />. Several generalizations of this theorem are known, see [[#References|[a1]]].

On the existence side, some families of Abelian difference sets are known, see [[#References|[a3]]].

==Examples.==
The most popular examples are as follows.

Cyclic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002042.png" />-difference sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002043.png" /> a prime power. The classical construction of these difference sets (elements in the multiplicative group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002044.png" /> whose trace is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002045.png" />) corresponds to the classical point-hyperplane designs of a finite projective space. For non-equivalent cyclic examples with the same parameters, see [[#References|[a5]]].

Quadratic residues in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002047.png" /> (Paley difference sets). Some other cyclotomic classes yield difference sets too, see [[#References|[a1]]].

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002048.png" />-difference sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002050.png" /> is a product of odd prime numbers (Hadamard difference sets, [[#References|[a2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002051.png" />, it is known that an Abelian Hadamard difference set exists if and only if the exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002052.png" /> is at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002053.png" />, see [[#References|[a4]]].

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002054.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002055.png" />-difference sets, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002057.png" /> an odd prime power) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002058.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002059.png" /> (generalized Hadamard difference sets, [[#References|[a2]]]).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002060.png" />-difference sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002061.png" /> a prime power (McFarland difference sets).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y.Q. Chen,   "On the existence of abelian Hadamard difference sets and generalized Hadamard difference sets"  ''Finite Fields and Appl.''  (to appear)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Jungnickel,   A. Pott,   "Difference sets: Abelian"  Ch.J. Colbourn (ed.)  J.H. Dinitz (ed.) , ''CRC Handbook of Combinatorial Designs'' , CRC  (1996)  pp. 297–307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.G. Kraemer,   "Proof of a conjecture on Hadamard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110020/a11002062.png" />-groups"  ''J. Combinatorial Th. A'' , '''63'''  (1993)  pp. 1–10</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Pott,   "Finite geometry and character theory" , ''Lecture Notes in Mathematics'' , '''1601''' , Springer  (1995)</TD></TR></table>