The Development of a Difference Set

Recall from the Difference Sets page that if $v$, $k$, and $\lambda$ are positive integers such that $2 \leq k < v$ and $(G, +)$ is a finite group of size $v$ with identity $0$, then a $(v, k, \lambda)$-difference set on $(G, +)$ is a subset $D \subseteq G$ such that $\mid D \mid = k$ and such that every element in $G \setminus \{ 0 \}$ is contained in the multiset $\{ x - y : x, y \in D \: \mathrm{and} \: x \neq y \}$ exactly $\lambda$ times.

Definition: Let $(G, +)$ be a finite abelian group containing a $(v, k, \lambda)$-difference set. For any element $g \in G$, the set $D + g = \{ x + g : x \in D \}$ is called a Translate of $D$.

A group $(G, +)$ is said to be abelian or commutative if for all $x, y \in G$ we have that $x + y = y + x$.

For example, consider the $(7, 3, 1)$-difference set $D = \{ 0, 1, 3 \}$ on the group $(\mathbb{Z}_7, +)$ where $\mathbb{Z}_7 = \{ 0, 1, 2, 3, 4, 5, 6 \}$. All of the possible translates of $D$ are:

(1)

\begin{align} \quad D + 0 = \{ 0, 1, 3 \} \\ \quad D + 1 = \{ 1, 2, 4 \} \\ \quad D + 2 = \{ 2, 3, 5 \} \\ \quad D + 3 = \{ 3, 4, 6 \} \\ \quad D + 4 = \{ 0, 4, 5 \} \\ \quad D + 5 = \{ 1, 5, 6 \} \\ \quad D + 6 = \{ 0, 2, 6 \} \end{align}

Since $(G, +)$ is a finite group and $D \subseteq G$ is a (finite) subset of $G$, the set of all translates of $D$ is also a finite set and is given an important name.

Definition: Let $(G, +)$ be a finite abelian group containing a $(v, k, \lambda)$-difference set. The Development of $D$ is the set of all translates of $D$ and is denoted $\mathrm{Dev} (D) = \{ D + g : g \in G \}$.

With the definitions made above we are able to take a difference set on a finite abelian group and obtain a symmetric BIBD.

Theorem 1: Let $(G, +)$ be a finite abelian group containing a $(v, k, \lambda)$-difference set $D$. Then $(G, \mathrm{Dev} (D))$ is a symmetric $(v, k, \lambda)$-BIBD.

• Proof: By definition we have that $\mid G \mid = v$.

• Furthermore, since $D$ contains $k$ elements, for any $g \in G$, the translate $D + g$ will also contain $k$ elements. It is clear that $D + g$ cannot contain more than $k$ elements, and if $D + g$ contained less than $k$ elements then this would mean that there exists elements $x, y \in D$ such that $x + g = y + g$. But then $(x + g) - g = (y + g) - g$, i.e., $x = y$, a contradiction. Thus every set in $\mathrm{Dev}(D)$ contains $k$ elements, i.e., every block contains $k$ points.

• Showing that every pair of distinct points $x, y \in G$ with $x \neq y$ is contained in $\lambda$ blocks (translates) is a bit more complicated so we will omit the proof for now.

• Lastly we show that this BIBD is symmetric by noting that $\mid \mathrm{Dev} (D) \mid = \mid G \mid = v$, i.e., $b = v$. $\blacksquare$