Abstract Algebra/Group Theory/Subgroup/Coset/a Group is Partitioned by Cosets of Its Subgroup

TheoremEdit

Let G be a Group. Let H be a Subgroup of G.

Then, Cosets of Subgroup H partition Group G.

ProofEdit

Overview: G is partition by the cosets if

  1. The cosets are subsets of G
  2. Each element of G is in one of the cosets.
  3. The cosets are disjoint

Cosets of H are Subsets of G Edit

0. Choose  
1. Choose  

By definition of gH

2.  

As Subgroup H is Subset of G

3.  

By 2., and Closure on G justified by 0. and 3.,

4.  

Each Element of G is in a Coset of H Edit

1.   subgroup inherits identity (usage 2)
2. Choose  
3  
definition of gH
4.  
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

2.  

By Definition of the Cosets,

3.  
Let  
4. Choose  

By Definition of g1H

5.  
6.  
7.  
8.  
9.  
10.  
11.  

As we can exchange g_1 and g_2 and apply the same procedure

12.  
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.