For questions about difference sets of groups, their developments to symmetric designs, their existence and non-existence, multipliers, and other properties.
A $(v,k,\lambda)$-difference set is a subset $D$ of a group $G$ such that $|D|=k$, $|G|=v$, and for all $g\in G$, there are exactly $\lambda$ ways to write $g=d_1d_2^{-1}$, where $d_1,d_2\in G$.
The term "difference set" comes about because for a group $G$ with group operation "$+$", the ways to write each $g\in G$ are literally differences of elements of $D$.
$D$ is said to inherit properties from $G$ such as being "abelian", "cyclic", etc. if $G$ has the respective property.
Difference sets yield symmetric designs in the following way:
Let $\text{dev}D$ denote the set $\lbrace D+g|g\in G\rbrace$. That is, $\text{dev}D$ (called the "development" of $D$) is the set of all blocks which are the right translates of $D$ by elements of $G$. It is easy to check that $\text{dev}D$ is in fact a symmetric $(v,k,\lambda)$-design.
For example, if $G=(\mathbb{Z}_7,+)$ with difference set $D=\lbrace 1,2,4 \rbrace$, then $\text{dev}D$ is the Fano plane. Note that $D$ is a $(7,3,1)$-difference set in this example. Because $\lambda=1$, the $(7,3,1)$-design $\text{dev}D$ is called planar (and is in fact a projective plane).
