Properties of Permutations
Every permutation can be written as a cycle or a product of disjoint cycles. This follows by considering the effect of a sequence of permutations on each member of the set
Each element i will end up after a sequence of permutations in some other position
meaning that somewhere is the simplified result, the sequence …, i, j, … must occur. There cannot be somewhere else a sequence of the form …, j, k, … or …, k, j, …since if this occurred, we would be able to simplify it.
Disjoint cycles commute. If
and
and the cycles have no element in common, then
Since
and
are disjoint we may write
where the
's are the symbols in neitheror
Suppose
is permuted by
then
sincefixes all theterms.
Similarly
so
for eachpermuted by
A similar argument is made for terms permuted by
and of course, bothandleave the
terms fixed. Hence
The order of a permutation of disjoint cycles is the least common multiple of the lengths of the cycles. To prove this, observe that a cycle of length
has order
Suppose then thatand are disjoint cycles of lengthand
and let
be the least common multiple ofand 
and
are the identity permutation, and sinceandcommute
Then the order of
must divide
Suppose the order ofis
then 
so
but ifandare disjoint, then so are
and
so both must be equal to
and both
anddividehence thatdivides
also and
The argument can be extended to any number of cycles in an obvious way.
Every permutation can be written in cycle form as a 2 cycle or a product of 2 cycles.
Note that
and
Permutations are odd or even. A permutation is even if it can be written as an even number of 2 cycles and odd if it can be written as an odd number of 2 cycles. Since every permutation can be written as a product of two cycles, the result follows.
The set of even permutations of
forms a subgroup of
To prove it, write each element
as a product of twocycles.
Define
then the kernel of
is the set of all even permutations.
The product of even permutations is even, as is the identity, and the inverse of an even permutation, so that the set of all even permutations is a subgroup.
