- Email: [email protected]
Group Divisible Difference Sets and Their Multtplers
Annals of Discrete Mathematics 8 (1980) 59-60
@ North-Holland Publishing Company.
GROUP DIVISIBLE DIFFERENCE SETS AND THEIR MULTIPLERS H.-P. KO Oakland University, Rochester, MI 48063, U.S.A.
D.K. RAY-CHAUDHURI Ohio State University, Columbus, OH 43210, U.S.A.
Abstract Let G be an abelian group of order mn and H be a subgroup of order n. A group divisible (GD) difference set for (G, H) with parameters (m,n, k, Al, A2) is a subset D c G with ID1 = k such that / DnD + X J
=
Al
if xeH\{O},
A2
if X E G \ H .
If H = {0} and h2= A, then a group divisible difference set is a (u, k, A)-difference set in the classical sense. Some of the principal contributors t o the theory of difference sets are Hall and Mann. Let u = mn and f be an integer such that (t, u) = 1. The integer t is called a multiplier of D iff for some group element g, tD = D + g. Difference sets and their multipliers is indeed one of the most elegant and deep subjects in combinatorics. Hall proved his beautiful multiplier theorem in 1936. We proved a general theorem on the subject of multipliers of group divisible difference sets and obtained many interesting results. This theorem contains theorems of Hall, Mann, Turyn, Hoffman, Jacobs and Elliott and Butson as special cases. A cyclic f i n e plane of order n is equivalent t o an (n + 1, n - 1, n, 0, 1)-DG difference set for g = Z n z P 1 and H = (n + l)G. We use the multiplier theorems to prove the non-existence of cyclic affine planes of order n, n ~ 5 , 0 0 0n, not a prime power.
59
@ North-Holland Publishing Company.
GROUP DIVISIBLE DIFFERENCE SETS AND THEIR MULTIPLERS H.-P. KO Oakland University, Rochester, MI 48063, U.S.A.
D.K. RAY-CHAUDHURI Ohio State University, Columbus, OH 43210, U.S.A.
Abstract Let G be an abelian group of order mn and H be a subgroup of order n. A group divisible (GD) difference set for (G, H) with parameters (m,n, k, Al, A2) is a subset D c G with ID1 = k such that / DnD + X J
=
Al
if xeH\{O},
A2
if X E G \ H .
If H = {0} and h2= A, then a group divisible difference set is a (u, k, A)-difference set in the classical sense. Some of the principal contributors t o the theory of difference sets are Hall and Mann. Let u = mn and f be an integer such that (t, u) = 1. The integer t is called a multiplier of D iff for some group element g, tD = D + g. Difference sets and their multipliers is indeed one of the most elegant and deep subjects in combinatorics. Hall proved his beautiful multiplier theorem in 1936. We proved a general theorem on the subject of multipliers of group divisible difference sets and obtained many interesting results. This theorem contains theorems of Hall, Mann, Turyn, Hoffman, Jacobs and Elliott and Butson as special cases. A cyclic f i n e plane of order n is equivalent t o an (n + 1, n - 1, n, 0, 1)-DG difference set for g = Z n z P 1 and H = (n + l)G. We use the multiplier theorems to prove the non-existence of cyclic affine planes of order n, n ~ 5 , 0 0 0n, not a prime power.
59