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

# Theorem

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

- Then, Cosets of Subgroup H partition Group G.

# Proof

editOverview: 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 *g*H

- 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

edit1. subgroup inherits identity (usage 2) 2. Choose - 3

definition of *g*H- 4.

*e*_{G}is identity of G (usage 3)

## The Cosets of H are Disjoint

edit- 0. Suppose 2 different cosets of H are not disjoint

- 1. Let the 2 cosets be
*g*_{1}H and*g*_{2}H where

Since they are not disjoint

- 2.

By Definition of the Cosets,

- 3.

- Let

- 4. Choose

By Definition of *g*_{1}H

- 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.