  A perfect isometry is an important relation between blocks of finite groups
as many information about blocks are preserved by it. If we consider the group
of all perfect isometries between a block to itself then this gives another
information about the block that is also preserved by a perfect isometry. The
structure of this group depends on the block and can be fairly simple or
extremely complicated. In this paper we study the perfect isometry group for
the block of $C_p$, the cyclic group of prime order, and completely describe
the structure of this group. The result shows that any self perfect isometry
for $C_p$ is essentially either induced by an element in $\mbox{Aut}(C_p)$, or
obtained by multiplication by one of its linear characters, or a composition of
both.
