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. _GX yields _{bar (G)}X, bar (G) being the homomorphic image of G in S_X, which was already mentioned. We also obtain the subactions   _GM on subsets M ÍX which are nonempty unions of orbits. Furthermore there are the restrictions  

of orbits of _UX, 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

• Double Cosets
• Action on k-subsets
• Sylows Theorem
• Products of Actions
• Paradigmatic Examples
• Paradigmatic Examples 2
• Exercises

Bilateral classes, symmetry classes of mappings