rdsshort.tex

Let $G$ be a finite group and $N\subseteq G$. The set $R\subseteq G$
with $|R|=k$ is called a ``relative difference set of order
$k-\lambda$ relative to the forbidden set $N$'' if the following
properties hold:

\beginlist%ordered{(a)}
\item{(a)} The multiset $\{ a.b^{-1}\colon a,b\in R\}$ contains
  every nontrivial ($\neq 1$) element of $G-N$ exactly $\lambda$
  times.  
\item{(b)} $\{ a.b^{-1}\colon a,b\in R\}$ does not contain
  any non-trivial element of $N$.
\endlist

Let $D\subseteq G$ be a difference set, then the incidence structure
with points $G$ and blocks $\{Dg\;|\;g\in G\}$ is called the
*development* of $D$. In short:  ${\rm dev} D$. Obviously, $G$ acts on
${\rm dev}D$ by multiplication from the right.

Relative difference sets with $N=1$ are called (ordinary) difference
sets. The development of a difference set with $N=1$ and $\lambda=1$
is projective plane of order $k-1$.

In group ring notation a relative difference set satisfies
$$
RR^{-1}=k+\lambda(G-N).
$$

The set $D\subseteq G$ is called *partial relative difference set*
with forbidden set $N$, if
$$
    DD^{-1}=\kappa+\sum_{g\in G-N}v_gg   
$$ 

holds for some $1\leq\kappa\leq k$ and $0\leq v_g \leq \lambda$ for
all $g\in G-N$.  If $D$ is a relative difference set then ,obviously,
$D$ is also a partial relative difference set.


*IMPORTANT NOTE*

\package{RDS} implicitly assumes that the *every* partial difference
set contains the identity element (see the notion of equivalence in
"RDS:Introduction" for the mathematical reason). However, the identity
*must not* be contained in the lists representing partial relative
difference sets.

So in \package{RDS}, the difference set `[ (), (1,2,3,4,5,6,7),
(1,4,7,3,6,2,5) ]' is represented by the list `[ (1,2,3,4,5,6,7),
(1,4,7,3,6,2,5) ]'. And no set of three non-trivial permutations will
be accepted as an ordinary difference set of `Group((1,2,3,4,5,6,7))'.

For this reason the lists returned by functions like "AllDiffsets" do
only contain non-trivial elements and look too short.



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