# Cosets math problem

1. Jan 13, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
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?
2. Relevant equations

3. The attempt at a solution

Last edited: Jan 13, 2008
2. Jan 13, 2008

### 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?

Last edited: Jan 13, 2008
3. Jan 13, 2008

### 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.