  In this thesis we study the following property of a finite group G: the
minimal number n such that G embeds in Sn. We start with an explicit formula
for the number n for abelian groups. Then, we study the behavior of this group
property in respect to direct products. Finally, we define and explore the
"compression ratio" of a finite group G which measures how much better the best
embedding is relative to the embedding given by Cayley's theorem.
