Consider an element 
in 
and assume that
are the conjugacy classes of 
. If
in standard cycle notation, then we associate with its 
-th cyclic factor
the element
of and call it the -th cycleproduct of or the
cycleproduct associated to
with respect to . In this way we obtain a total of
cycleproducts,
of them arising from the cyclic factors of 
which are of length 
. Now let 
be the number of these
cycleproducts which are associated to a -cycle of and which belong
to the conjugacy class 
of (note that we did not
say
let be the number of different cycleproducts ).
We
put these natural numbers together into the matrix
This matrix has 
columns ( is the column index)
and as many rows as there are conjugacy classes
in (
is the row index). Its entries satisfy the following conditions:
We call this matrix 
the type
of and we say that is of type .
occurs as the type of an element
where 
.
Lemma
The conjugacy classes of
complete monomial groups have the following properties:
.
in
, finite, is equal to
with columns and as many rows as has
conjugacy classes, the elements of which satisfy
.
is a permutation group and 
, then the cycle partition 
, where
denotes the permutation representation of 
,
is equal
to
, is defined to be
, and where 
means that the proper partition has to be formed that consists of all the
parts of all the summands 
.
Proof: A first remark concerns the cycleproducts introduced in . Since
in each group 
the products 
and 
of two elements are conjugate, we
have that
is conjugate to
for each integer 
.
The second remark is, that for each 
and every 
,
This follows from the fact that both 
and
are of type .
A third remark is that 
implies the existence of an
element 
which satisfies 
, and for which
the cycleproducts and 
are
conjugate.
It is not difficult to check these remarks and then to derive the statement
(exercise ).
Exercises
E 
.
Prove that the conjugacy class (in 
) of an even element 
splits into two 
-classes if and only if
the lengths of the
cyclic factors of are pairwise different and odd (hint: use ).
E 
.
Fill in the details of the proof of .
