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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 Sym(S) 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.
* Under a group action 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 contradiction); 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
is 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.
Next: Simplicity of the Alternating
Up: No Title
Previous: The isomorphism theorems
Gregory Constantine
1998-09-01