Cosets of Subgroups: Is Each Class a Group?

[ehrenfest]

Homework Statement

Let H be a subgroup of a finite group G. I understand that the cosets of H partition G into equivalence classes. Is it always true that each of these equivalence classes is a group?

EDIT: clearly is it not always true; let H ={0,4,8,12} in Z_16 and take the right coset with 1; so are there conditions that make it true?

Homework Equations

The Attempt at a Solution

 

[Mathdope]

I think the cosets form a group iff the subgroup H is normal.

Edit: sorry I think I misread; are you asking if each coset forms a group or if the collection of cosets forms a group?

 

[Dick]

Only one coset can be a group. The one containing e. The quotient group (collection of cosets) can be a group as Mathdope alluded to.