# Abelian difference set

Let $ G $
be a group of order $ v $
and $ D \subseteq G $
with $ | D | = k $.
Then $ D $
is called a $ ( v,k, \lambda ) $-
difference set of order $ n = k - \lambda $
in $ G $
if every element $ g \neq 1 $
in $ G $
has exactly $ \lambda $
different representations $ g = d \cdot d ^ {\prime - 1 } $
with $ d,d ^ \prime \in D $,
see [a1]. For instance, $ \{ 1,2,4 \} $
is a $ ( 7,3,1 ) $-
difference set in the cyclic group of order $ 7 $.
If $ G $
is Abelian (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets $ D _ {1} $
and $ D _ {2} $
in $ G $
are equivalent if there is a group automorphism $ \varphi $
such that $ \varphi ( D _ {1} ) = D _ {2} g $.
The existence of a $ ( v,k, \lambda ) $-
difference set is equivalent to the existence of a symmetric $ ( v,k, \lambda ) $-
design with $ G $
acting as a regular automorphism group (cf. also Difference set). If two difference sets correspond to isomorphic designs, the difference sets are called isomorphic. Given certain parameters $ v $,
$ k $
and $ \lambda $
and a group $ G $,
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 are required: The existence of the difference set implies the existence of an algebraic integer of absolute value $ n $
in some cyclotomic field. In several cases one can prove that no such element exists, see [a5]. Another approach for non-existence results uses multipliers: A multiplier of an Abelian difference set in $ G $
is an automorphism $ \varphi $
of $ G $
such that $ \varphi ( D ) = Dg $.
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 $ g \mapsto g ^ {t} $
is a multiplier of an Abelian difference set provided that: i) $ t $
divides the order $ n $;
ii) $ t $
is relatively prime to $ v $;
and iii) $ t > \lambda $.
Several generalizations of this theorem are known, see [a1].

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

## Examples.

The most popular examples are as follows.

Cyclic $ \left ( { \frac{q ^ {d + 1 } - 1 }{q - 1 } } , { \frac{q ^ {d} - 1 }{q - 1 } } , { \frac{q ^ {d - 1 } - 1 }{q - 1 } } \right ) $- difference sets, $ q $ a prime power. The classical construction of these difference sets (elements in the multiplicative group of $ { \mathop{\rm GF} } ( q ^ {d + 1 } ) $ whose trace is $ 0 $) corresponds to the classical point-hyperplane designs of a finite projective space. For non-equivalent cyclic examples with the same parameters, see [a5].

Quadratic residues in $ { \mathop{\rm GF} } ( q ) $, $ q \equiv 3 ( { \mathop{\rm mod} } 4 ) $( Paley difference sets). Some other cyclotomic classes yield difference sets too, see [a1].

$ ( 4m ^ {2} , 2m ^ {2} - m, m ^ {2} - m ) $- difference sets, $ m = 2 ^ {a} 3 ^ {b} u ^ {2} $, where $ u $ is a product of odd prime numbers (Hadamard difference sets, [a2]). If $ m = 2 ^ {a} $, it is known that an Abelian Hadamard difference set exists if and only if the exponent of $ G $ is at most $ 2 ^ {a + 2 } $, see [a4].

$ \left ( 4m ^ {2n } \cdot { \frac{m ^ {2n } - 1 }{m ^ {2} - 1 } } , m ^ {2n - 1 } \cdot \left ( { \frac{2 ( m ^ {2n } - 1 ) }{m + 1 } } + 1 \right ) , \right . $ $ \left . ( m ^ {2n } - m ^ {2n - 1 } ) \cdot { \frac{m ^ {2n - 1 } + 1 }{m + 1 } } \right ) $- difference sets, where $ m = q ^ {2} $( $ q $ an odd prime power) or $ m = 3 ^ {t} $ or $ m =2 $( generalized Hadamard difference sets, [a2]).

$ \left ( q ^ {d + 1 } \left ( 1 + { \frac{q ^ {d + 1 } - 1 }{q - 1 } } \right ) , q ^ {d} \cdot { \frac{q ^ {d + 1 } - 1 }{q - 1 } } , q ^ {d} \cdot { \frac{q ^ {d} - 1 }{q - 1 } } \right ) $- difference sets, $ q $ a prime power (McFarland difference sets).

#### References

[a1] | T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1986) |

[a2] | Y.Q. Chen, "On the existence of abelian Hadamard difference sets and generalized Hadamard difference sets" Finite Fields and Appl. (to appear) |

[a3] | 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 |

[a4] | R.G. Kraemer, "Proof of a conjecture on Hadamard $2$-groups" J. Combinatorial Th. A , 63 (1993) pp. 1–10 |

[a5] | A. Pott, "Finite geometry and character theory" , Lecture Notes in Mathematics , 1601 , Springer (1995) |

**How to Cite This Entry:**

Abelian difference set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Abelian_difference_set&oldid=53241