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Multiplier theorems

✍ Scribed by K. T. Arasu; Qing Xiang

Publisher
John Wiley and Sons
Year
1995
Tongue
English
Weight
516 KB
Volume
3
Category
Article
ISSN
1063-8539

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✦ Synopsis

We state and prove a multiplier theorem for a central element A of ZG, the group ring over Z of a group G. This generalizes most previously known multiplier theorems for difference sets and divisible difference sets. We also provide applications to show that our theorem provides new multipliers and establish the nonexistence of a family of divisible difference sets which correspond to elliptic semiplanes admitting a regular collineation group. o 1995 John Wiley & Sons, Inc.

k(k -1) = A(v -1).

We define n = k -A; n is called the order of D . We say that D is cyclic (resp. abelian) if G is cyclic (resp. abelian).

Let R be a commutative ring with unity 1 and G be a group. Let RG denote the group ring of G over R. We identify each subset S of G with the group ring element S = EXES x .

For A = EgEG a,g E RG and any integer t , we define A(') = EgEG a,g'.

With these notations, the difference set condition of D C G can be translated into

DD(-')

= n + AG in ZG. where n = n 1 in ZG

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