Essential Concepts of Cosets

Cosets are essential in understanding group structures by grouping elements based on subgroups. They help partition a group into distinct subsets, revealing relationships between groups and their subgroups, which is crucial in the study of Groups and Geometries.

1. Definition of a coset

   • A coset is a form of grouping elements of a group based on a subgroup.
   • Given a group ( G ) and a subgroup ( H ), the left coset of ( H ) with respect to an element ( g \in G ) is ( gH = { gh \mid h \in H } ).
   • The right coset is defined similarly as ( Hg = { hg \mid h \in H } ).
   • Cosets partition the group into disjoint subsets.

2. Left cosets

   • Left cosets are formed by multiplying each element of the subgroup ( H ) by a fixed element ( g ) from the group ( G ).
   • All left cosets of ( H ) in ( G ) have the same number of elements as ( H ).
   • If ( g_1H = g_2H ), then ( g_1 ) and ( g_2 ) are in the same left coset.

3. Right cosets

   • Right cosets are formed by multiplying each element of the subgroup ( H ) by a fixed element ( g ) from the group ( G ) on the right: ( Hg = { hg \mid h \in H } ).
   • Similar to left cosets, all right cosets of ( H ) in ( G ) have the same number of elements as ( H ).
   • If ( Hg_1 = Hg_2 ), then ( g_1 ) and ( g_2 ) are in the same right coset.

4. Coset representatives

   • A coset representative is a single element chosen from each coset of a subgroup.
   • Each coset can be uniquely represented by one of its elements.
   • The choice of representatives can simplify calculations and proofs involving cosets.

5. Index of a subgroup

   • The index of a subgroup ( H ) in a group ( G ), denoted ( [G : H] ), is the number of distinct cosets of ( H ) in ( G ).
   • It is calculated as ( |G| / |H| ) when both groups are finite.
   • The index provides insight into the structure of the group and the subgroup.

6. Lagrange's Theorem

   • Lagrange's Theorem states that the order (number of elements) of a subgroup ( H ) divides the order of the group ( G ).
   • It implies that the index ( [G : H] ) is an integer.
   • The theorem is fundamental in understanding the relationship between groups and their subgroups.

7. Coset multiplication

   • Coset multiplication involves the operation of cosets, where the product of two cosets is defined if they are compatible.
   • For left cosets, ( g_1H \cdot g_2H = g_1g_2H ) if ( g_1H ) and ( g_2H ) are defined.
   • This operation is well-defined if ( H ) is a normal subgroup.

8. Normal subgroups and cosets

   • A subgroup ( H ) is normal in ( G ) if ( gH = Hg ) for all ( g \in G ).
   • Normal subgroups allow for the formation of quotient groups, where cosets can be treated as single elements.
   • The concept of normality is crucial for understanding the structure of groups.

9. Quotient groups

   • A quotient group ( G/H ) is formed by the set of cosets of a normal subgroup ( H ) in ( G ).
   • The operation on the quotient group is defined by the multiplication of cosets.
   • Quotient groups help in analyzing the structure of groups and their subgroups.

10. Coset enumeration

    • Coset enumeration is the process of counting the number of distinct cosets of a subgroup in a group.
    • It is useful for understanding the structure and size of groups.
    • Techniques such as the orbit-stabilizer theorem can aid in coset enumeration.