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 The cosets are subsets of G Each element of G is in one of the cosets. The cosets are disjoint Sienna Cosets of H are Subsets of G 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. Sienna Each Element of G is in a Coset of H {| Style ="width:60%" |- | 1. || subgroup inherits identity (usage 2) |- | 2. Choose || |- | 3 || definition of gH |- | 4. || eG is identity of G (usage 3) |} Sienna The Cosets of H are Disjoint 0. Suppose 2 different cosets of H are not disjoint 1. Let the 2 cosets be and 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.