| | | **Bilateral classes, symmetry classes of mappings** |

## Bilateral classes, symmetry classes of mappings

From a given action we can
derive various other actions in a natural
way, e.g. _{G}X yields _{ bar (G)}X, * bar (G)* being the
homomorphic image of *G* in *S*_{X}, which was already mentioned. We also
obtain the *subactions*
_{G}M on subsets *M ÍX* which
are nonempty unions of orbits. Furthermore there are the *restrictions*
_{U}X to the subgroups *U* of *G*. As the orbits of _{G}X are unions
of orbits of _{U}X, the comparison of actions and restrictions is a
suitable way of generalizing or specializing structures if they can be
defined as orbits. The following example will show what is meant by this.
**Examples: **
Let *U* denote a subgroup of the direct product
*G ´G*. Then U acts on *G* as follows:
*U ´G -> G :((a,b),g) -> agb*^{-1}.

The orbits *U(g)= {agb*^{-1} | (a,b) ÎU } of this action are called
the *bilateral classes*
of *G* with respect to *U*. By specializing
*U* we obtain various interesting group theoretical structures some of
which have been mentioned already:
- If
*A* is a subgroup of *G*, then both *A ´{1 }* and * {1 }
´A* are subgroups of *G ´G*. Their orbits are the subsets
*(A ´{1 })(g)=Ag,
*

the *right cosets*
of *A* in *G*, and
*( {1 } ´A)(g)=gA,
*

the *left cosets* of *A* in *G*.
- If
*B* denotes a second subgroup
of *G*, then we can put *U* equal to the subgroup *A ´B*,
obtaining as orbits
the
*(A,B)*- *double cosets*
of *G*:
*(A ´B)(g)=AgB.
*

- Another subgroup of
*G ´G* is its *diagonal*
subgroup
* D(G ´G)
:= {(g,g) | g ÎG }.
*

Its orbits are the conjugacy classes:
* D(G ´G)(g)= {g'gg'*^{-1} | g' ÎG }=C^{G}(g).

Hence left and right cosets, double cosets and conjugacy classes turn out to
be special cases of bilateral classes. Being orbits, two of them are either equal or
disjoint, moreover, their order is the index of the stabilizer of an element.
We have mentioned this in connection with conjugacy classes and centralizers
of elements, here is the consequence for double cosets: Since

*(A ´B)*_{g}= {(gbg^{-1},b) | b ÎB },

we obtain

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

| | | **Bilateral classes, symmetry classes of mappings** |