In combinatorics, a ${\displaystyle (v,k,\lambda )}$ difference set is a subset ${\displaystyle D}$ of size ${\displaystyle k}$ of a group ${\displaystyle G}$ of order ${\displaystyle v}$ such that every non-identity element of ${\displaystyle G}$ can be expressed as a product ${\displaystyle d_{1}d_{2}^{-1}}$ of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. A difference set ${\displaystyle D}$ is said to be cyclic, abelian, non-abelian, etc., if the group ${\displaystyle G}$ has the corresponding property. A difference set with ${\displaystyle \lambda =1}$ is sometimes called planar or simple.^[1] If ${\displaystyle G}$ is an abelian group written in additive notation, the defining condition is that every non-zero element of ${\displaystyle G}$ can be written as a difference of elements of ${\displaystyle D}$ in exactly ${\displaystyle \lambda }$ ways. The term "difference set" arises in this way.

Basic facts

• A simple counting argument shows that there are exactly ${\displaystyle k^{2}-k}$ pairs of elements from ${\displaystyle D}$ that will yield nonidentity elements, so every difference set must satisfy the equation ${\displaystyle k^{2}-k=(v-1)\lambda .}$
• If ${\displaystyle D}$ is a difference set and ${\displaystyle g\in G,}$ then ${\displaystyle gD=\{gd:d\in D\}}$ is also a difference set, and is called a translate of ${\displaystyle D}$ (${\displaystyle D+g}$ in additive notation).
• The complement of a ${\displaystyle (v,k,\lambda )}$-difference set is a ${\displaystyle (v,v-k,v-2k+\lambda )}$-difference set.^[2]
• The set of all translates of a difference set ${\displaystyle D}$ forms a symmetric block design, called the development of ${\displaystyle D}$ and denoted by ${\displaystyle dev(D).}$ In such a design there are ${\displaystyle v}$ elements (usually called points) and ${\displaystyle v}$ blocks (subsets). Each block of the design consists of ${\displaystyle k}$ points, each point is contained in ${\displaystyle k}$ blocks. Any two blocks have exactly ${\displaystyle \lambda }$ elements in common and any two points are simultaneously contained in exactly ${\displaystyle \lambda }$ blocks. The group ${\displaystyle G}$ acts as an automorphism group of the design. It is sharply transitive on both points and blocks.^[3]
  • In particular, if ${\displaystyle \lambda =1}$, then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group ${\displaystyle \mathbb {Z} /7\mathbb {Z} }$ is the subset ${\displaystyle \{1,2,4\}}$. The translates of this difference set form the Fano plane.
• Since every difference set gives a symmetric design, the parameter set must satisfy the Bruck–Ryser–Chowla theorem.^[4]
• Not every symmetric design gives a difference set.^[5]

Equivalent and isomorphic difference sets

Two difference sets ${\displaystyle D_{1}}$ in group ${\displaystyle G_{1}}$ and ${\displaystyle D_{2}}$ in group ${\displaystyle G_{2}}$ are equivalent if there is a group isomorphism ${\displaystyle \psi }$ between ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ such that ${\displaystyle D_{1}^{\psi }=\{d^{\psi }\colon d\in D_{1}\}=gD_{2}}$ for some ${\displaystyle g\in G_{2}.}$ The two difference sets are isomorphic if the designs ${\displaystyle dev(D_{1})}$ and ${\displaystyle dev(D_{2})}$ are isomorphic as block designs.

Equivalent difference sets are isomorphic, but there exist examples of isomorphic difference sets which are not equivalent. In the cyclic difference set case, all known isomorphic difference sets are equivalent.^[6]

Multipliers

A multiplier of a difference set ${\displaystyle D}$ in group ${\displaystyle G}$ is a group automorphism ${\displaystyle \phi }$ of ${\displaystyle G}$ such that ${\displaystyle D^{\phi }=gD}$ for some ${\displaystyle g\in G.}$ If ${\displaystyle G}$Zdroj:https://en.wikipedia.org?pojem=Difference_set
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.