#### Title

Generalized Addition Sets

#### Date of Award

1980

#### Availability

Article

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics

#### Abstract

An (m,n,k,(lamda),(lamda)',g) relative addition set, which is a generalization of an addition set and a relative difference set, is defined and studied. It has properties similar to an addition set and a relative difference set. It is shown that there are essentially no nontrivial (m,n,k,(lamda),(lamda)',l) relative addition sets with mn odd. The connections among addition sets, relative addition sets and block designs are studied and it is shown that an addition set and a relative addition set both imply the existence of a partially balanced incomplete block design of a certain order. As a result, a generalization of a well-known difference set equation is obtained. A (v,k,(lamda),(alpha)) group addition set, in a finite group G, is defined where (alpha): G (--->) G is a homomorphism. Many of the same properties of addition sets hold for abelian group addition sets. It is shown that there is no abelian group addition set with v odd or v (TBOND) 2 (mod 4), and (alpha)= I, where I is the identity automorphism. Examples of both abelian and nonabelian group addition sets are given.

#### Keywords

Mathematics

#### Recommended Citation

Sumner, John Sanford, "Generalized Addition Sets" (1980). *Dissertations from ProQuest*. 1129.

https://scholarlyrepository.miami.edu/dissertations/1129

## Link to Full Text

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