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

Theorem

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

Then, Cosets of Subgroup H partition Group G.

Proof

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

0. Choose $g\in G$

1. Choose $k\in gH$

By definition of gH

2. $\exists \;h\in H:k=g\ast h$

As Subgroup H is Subset of G

3. $h\in G$

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

4. $k=g\ast h\in G$

Each Element of G is in a Coset of H

1. $e_{G}\in H$	subgroup inherits identity (usage 2)
2. Choose $g\in G$
3 $g\ast e_{G}\in gH$	definition of gH
4. $g=g\ast e_{G}\in gH$	eG is identity of G (usage 3)

The Cosets of H are Disjoint

0. Suppose 2 different cosets of H are not disjoint

1. Let the 2 cosets be g₁H and g₂H where ${\color {Blue}g_{1}},{\color {OliveGreen}g_{2}}\in G$

Since they are not disjoint

2. $\exists u\in G:u\in {\color {Blue}g_{1}}H{\text{ and }}u\in {\color {OliveGreen}g_{2}}H$

By Definition of the Cosets,

3. $\exists {\color {Blue}h_{1}},{\color {OliveGreen}h_{2}}\in H:{\color {Blue}g_{1}}\ast {\color {Blue}h_{1}}=u={\color {OliveGreen}g_{2}}\ast {\color {OliveGreen}h_{2}}$

Let $z={\color {OliveGreen}g_{2}^{-1}}\ast {\color {Blue}g_{1}}={\color {OliveGreen}h_{2}}\ast {\color {Blue}h_{1}^{-1}}\in H$

4. Choose $k\in g_{1}H$

By Definition of g₁H

5. $\exists \;h_{k}\in H:k=g_{1}\ast h_{k}$

6. $z\ast h_{k}\in H$

7. $g_{2}\ast (z\ast h_{k})\in g_{2}H$

8. $g_{2}\ast g_{2}^{-1}\ast g_{1}\ast h_{k}\in g_{2}H$

9. $g_{1}\ast h_{k}\in g_{2}H$

10. $k\in g_{2}H$

11. $g_{1}H\subseteq g_{2}H$

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

12. $g_{2}H\subseteq g_{1}H$

13. $g_{1}H=g_{2}H$ 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.