Conjugation
Having described the elements of Sn , we show which
of them are in the same conjugacy class, i.e. in the same
orbit of the group Sn
on the set Sn under the
conjugation action. In order to do this, we first
note how rpr-1 is obtained from the permutation p:
Thus, in terms of cyclic factors of p, rpr-1 arises from
p = ( ...i pi ...) ...
by simply
applying r to the points in the cycles of p:
rpr-1= ...( ...ri r( pi) ...)
... .
(For any mapping j:S -> T we mean by j(s)=t
that s has to
be replaced by t and not that t is replaced by s,
as it is sometimes understood!)
This equation shows that the lengths of the cyclic factors of p
are the same as those of rpr-1. It is easy to see that,
conversely, for any two elements p, sÎSn with the same
lengths l n of cyclic factors there exists a rÎSn
such that rpr-1= s. Hence the lengths of
the cyclic factors of p characterize its conjugacy class.
To make this more explicit, we introduce the notion of (proper) partition
of n Î N, by which we mean any sequence a= ( a1, a2,
...) of natural numbers ai which satisfy
" i : ai >= ai+1, and åi ai=n.
The ai are called the parts
of a. The fact that a
is a partition of n is abbreviated by
a|¾n.
If a|¾n then there exists an h such that ai=0 for
all i>h. We may therefore write
a=( a1, ..., ah),
for any such h. The minimal h with this property will be denoted
by l( a) and called the length
of a.
The following abbreviation is useful in the case when several
nonzero parts of
a are equal, say ai parts are equal to i,i În:
a= (nan,(n-1)an-1, ...,1a1).
If ai=0, then iai is usually omitted, e.g.
(3,12)=(3,1,1,0, ...).
For pÎSn the ordered lengths ai( p),i Î c( p),
of the cyclic factors of p in cycle notation form a uniquely determined
proper partition
a( p)=( a1( p), a2( p), ..., ac( p)
( p)) |¾n,
which we call the cycle partition
of p. The corresponding
n-tuple
a( p):=(a1( p), ...,an( p))
consisting of the multiplicities ai( p) of the parts of length i in
a( p)
is called the cycle type
of p. Correspondingly we call an n-tuple
a:=(a1, ...,an) a
cycle type of n if and only if
each ai Î N, and
åi ·ai
=n.
This will be abbreviated by
a |¾| n.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
