Difference between revisions of "Orbit"
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− | An '''orbit''' is part of a [[set]] on which a [[group]] acts. | + | An '''orbit''' is part of a [[set]] on which a [[group]] [[group action|acts]]. |
Let <math>G</math> be a group, and let <math>S</math> be a <math>G</math>-set. The '''orbit''' of an element <math>x\in S</math> is the set <math>Gx</math>, i.e., the set of [[conjugate (group theory) | conjugate]]s of <math>x</math>, or the set of elements <math>y</math> in <math>S</math> for which there exists <math>\alpha \in G</math> for which <math>\alpha x = y</math>. | Let <math>G</math> be a group, and let <math>S</math> be a <math>G</math>-set. The '''orbit''' of an element <math>x\in S</math> is the set <math>Gx</math>, i.e., the set of [[conjugate (group theory) | conjugate]]s of <math>x</math>, or the set of elements <math>y</math> in <math>S</math> for which there exists <math>\alpha \in G</math> for which <math>\alpha x = y</math>. | ||
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* [[Stabilizer]] | * [[Stabilizer]] | ||
* [[Orbit-stabilizer theorem]] | * [[Orbit-stabilizer theorem]] | ||
+ | * [[Group action]] | ||
[[Category:Group theory]] | [[Category:Group theory]] |
Revision as of 16:33, 7 September 2008
An orbit is part of a set on which a group acts.
Let be a group, and let
be a
-set. The orbit of an element
is the set
, i.e., the set of conjugates of
, or the set of elements
in
for which there exists
for which
.
For , the mapping
is sometimes known as the orbital mapping defined by
; it is a homomorphism of the
-set
(with action on itself, by left translation) into
; the image of
is the orbit of
. We say that
acts freely on
if the orbital mapping defined by
is injective, for all
.
The set of orbits of is the quotient set of
under the relation of conjugation. This set is denoted
, or
. (Sometimes the first notation is used when
acts on the left, and the second, when
acts on the right.)
Let be a set acting on
from the right, and let
be a normal subgroup of
. Then
acts on
from the right, under the action
, for
. (
acts trivially on this set, so
.) Consider the canonical mapping
. The inverse images of elements of
under
are the orbits of
under action of
; thus on passing to the quotient,
defines an isomorphism from
to
.
Suppose and
are groups, and
acts on
on the left, and
on the right; suppose furthermore that the operations of
and
commute, i.e., for all
,
,
,
Let
be the opposite group of
; then the actions of
and
on
define a left action of
on
. The set
is denoted
. Since
and
are normal subgroups of
, by the previous paragraph, the
-sets
,
,
are isomorphic and identitfied with each other.
Let be a group, and
a subgroup of
; let it act on
from the right. Then the set
is the set of left cosets mod
.
If is a group and
are subgroups of
, then the set
is called the set of double cosets mod
and
.