Group actions
Let S be a set. A bijection
is called a permutation. Let
permutations on S
If
and
then
denotes the
image of
under f. The set
forms a group under composition of
mappings, called the symmetric group on S.
We say that the group G acts on the set S if there is a
homomorphism
For
in S and g in G we abbreviate by
writing
in place of
if h is also in G we have
.
If kerT=1 we say that G
acts faithfully.
Define a subgroup
and call it the stabilizer of
in G. Further define
and call this subset of S
the orbit of
under the action of G.
* The set S decomposes into a disjoint union of orbits.
Indeed, pick an
element
of S and produce its orbit under the action of G; pick
another element
of S outside this orbit and produce its orbit (the
two orbits are easily seen to be disjoint, since
implies
a cotradiction); proceed until S is exhausted.
* The cardinality (or length) of the orbit is equal to the index of
the stabilizer.
Specifically, we assert that
This follows by
observing that
is sent into
by exacly those group elements that
are in the coset
.
* (The class equation)
where
is a representative
from orbit i.
Indeed, S is the disjoint union of orbits. Its cardinality is, therefore,
the sum of the lengths of these orbits, which are indices of stabilizers
of orbit representatives.
* The stabilizers of two elements from the same orbit are conjugate
in G. [Specifically,
]
The map
establishes a
bijection between
and
,
which allows us to conclude that
The homomorphic image T(G) is called a permutation representation
of the group G on the set S. We call T(G) a permutation group on S.
A permutation group on a set S is called transitive if for
any two elements of S there exists an element of the group sending
one into the other.
When the group element g is viewed as a permutation, the elements
of S that it fixes are called the fixed points of g.
A transitive group is regular if the only element of the
group which has fixed points is the identity.
Gregory Constantine
1998-07-14