# 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 ${\displaystyle g\in G}$
1. Choose ${\displaystyle k\in gH}$

By definition of gH

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

As Subgroup H is Subset of G

3. ${\displaystyle h\in G}$

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

4. ${\displaystyle k=g\ast h\in G}$

## Each Element of G is in a Coset of H

 1. ${\displaystyle e_{G}\in H}$ subgroup inherits identity (usage 2) 2. Choose ${\displaystyle g\in G}$ 3 ${\displaystyle g\ast e_{G}\in gH}$ definition of gH 4. ${\displaystyle g=g\ast e_{G}\in gH}$ 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 ${\displaystyle {\color {Blue}g_{1}},{\color {OliveGreen}g_{2}}\in G}$

Since they are not disjoint

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

By Definition of the Cosets,

3. ${\displaystyle \exists {\color {Blue}h_{1}},{\color {OliveGreen}h_{2}}\in H:{\color {Blue}g_{1}}\ast {\color {Blue}h_{1}}=u={\color {OliveGreen}g_{2}}\ast {\color {OliveGreen}h_{2}}}$
Let ${\displaystyle z={\color {OliveGreen}g_{2}^{-1}}\ast {\color {Blue}g_{1}}={\color {OliveGreen}h_{2}}\ast {\color {Blue}h_{1}^{-1}}\in H}$
4. Choose ${\displaystyle k\in g_{1}H}$

By Definition of g1H

5. ${\displaystyle \exists \;h_{k}\in H:k=g_{1}\ast h_{k}}$
6. ${\displaystyle z\ast h_{k}\in H}$
7. ${\displaystyle g_{2}\ast (z\ast h_{k})\in g_{2}H}$
8. ${\displaystyle g_{2}\ast g_{2}^{-1}\ast g_{1}\ast h_{k}\in g_{2}H}$
9. ${\displaystyle g_{1}\ast h_{k}\in g_{2}H}$
10. ${\displaystyle k\in g_{2}H}$
11. ${\displaystyle g_{1}H\subseteq g_{2}H}$

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

12. ${\displaystyle g_{2}H\subseteq g_{1}H}$
13. ${\displaystyle g_{1}H=g_{2}H}$  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.