Having described the elements of 
, we show which
of them are in the same conjugacy class, i.e. in the same
orbit of the group
on the set under the
conjugation action (cf. 
). In order to do this, we first
note how 
is obtained from the permutation 
:
Thus, in terms of cyclic factors of , arises from
by simply
applying 
to the points in the cycles of :
(For any mapping 
we mean by 
that 
has to
be replaced by 
and not that is replaced by 
as it is sometimes understood!)
This equation shows that the lengths of the cyclic factors of
are the same as those of . It is easy to see that,
conversely, for any two elements 
with the same
lengths 
of cyclic factors there exists a 
such that 
. Hence the lengths of
the cyclic factors of characterize its conjugacy class.
To make this more explicit, we introduce the notion of (proper) partition
of 
, by which we mean any sequence 
of natural numbers 
which satisfy
The 
are called the parts
of 
. The fact that 
is a partition of 
is abbreviated by
If 
then there exists an 
such that 
for
all 
. We may therefore write
for any such . The minimal with this property will be denoted
by 
and called the length
of 
.
The following abbreviation is useful in the case when several
nonzero parts of
are equal, say 
parts are equal to 
:
If 
, then 
is usually omitted, e.g.
For 
the ordered lengths 
,
of the cyclic factors of in cycle notation form a uniquely determined
proper partition
which we call the cycle partition
of . The corresponding
-tuple
consisting of the multiplicities 
of the parts of length 
in
is called the cycle type
of . Correspondingly we call an -tuple
a
cycle type of if and only if
each 
, and
This will be abbreviated by
