Abstract Algebra/Group Theory/Subgroup/Coset/a Group is Partitioned by Cosets of Its Subgroup
Let G be a Group. Let H be a Subgroup of G.
- Then, Cosets of Subgroup H partition Group G.
Overview: G is partition by the cosets if
- The cosets are subsets of G
- Each element of G is in one of the cosets.
- The cosets are disjoint
Cosets of H are Subsets of G Edit
- 0. Choose
- 1. Choose
By definition of gH
As Subgroup H is Subset of G
By 2., and Closure on G justified by 0. and 3.,
Each Element of G is in a Coset of H Edit
1. subgroup inherits identity (usage 2) 2. Choose
definition of gH
eG is identity of G (usage 3)
The Cosets of H are DisjointEdit
- 0. Suppose 2 different cosets of H are not disjoint
- 1. Let the 2 cosets be g1H and g2H where
Since they are not disjoint
By Definition of the Cosets,
- 4. Choose
By Definition of g1H
As we can exchange g_1 and g_2 and apply the same procedure
- 13. contradicting that the two coset are different (0.)
Thus, two Cosets of H are either identical or are disjoint. Hence, the Cosets of H are disjoint.