-
Let
. The left coset of
containing is defined as and the right coset of
containing is defined as -
(Fraleigh 10.1) The cosets partition a group into equivalence classes.
-
More specifically the left coset expresses the equivalence relation
-
The right coset expresses
-
-
(Fraleigh e10.34) There are the same number of left and right cosets of a subgroup
. That is, we can establish a one-to-one map between left and right cosets. - Intuition: Note that
and that the inverse is unique. Map to
- Intuition: Note that
-
Every coset of a subgroup
has the same number of elements as . -
Let
. The double coset is the set defined as
- This forms an equivalence class where
if and only if there exists and such that .
- This forms an equivalence class where
Group Index
- Let
, the number of left cosets of in is called the index of and . This is denoted . - (Fraleigh 10.14) Let
such that and suppose and are both finite. Then is finite and