- #1
murmillo
- 118
- 0
1. The problem statement, all variables and given/known
I am reading about cosets and am stuck on this proposition. Let H be a subgroup of a group G. If aH and bH have an element in common, then they are equal.
But let the group be Z with addition as the law of composition. Let H be 5Z, the set of all multiples of 5. Then 4H and 12H are cosets, and 60 is an element of 4H (since 60=4(15) and 15 is in H). But 60 is also an element of 12H (since 60=12(5) and 5 is in H). But according to the proposition, 4H and 12H must then be equal, but they are not, since 20 is an element of 4H but not 12H. What am I doing wrong?
aH = {ah where h is an element of H)
I thought that part of the problem has to do with my using multiplication, but I don't think that's right. I've read through the section on cosets several times but I still can't figure out what I'm doing wrong. I'm sure it's something obvious.
I am reading about cosets and am stuck on this proposition. Let H be a subgroup of a group G. If aH and bH have an element in common, then they are equal.
But let the group be Z with addition as the law of composition. Let H be 5Z, the set of all multiples of 5. Then 4H and 12H are cosets, and 60 is an element of 4H (since 60=4(15) and 15 is in H). But 60 is also an element of 12H (since 60=12(5) and 5 is in H). But according to the proposition, 4H and 12H must then be equal, but they are not, since 20 is an element of 4H but not 12H. What am I doing wrong?
Homework Equations
aH = {ah where h is an element of H)
The Attempt at a Solution
I thought that part of the problem has to do with my using multiplication, but I don't think that's right. I've read through the section on cosets several times but I still can't figure out what I'm doing wrong. I'm sure it's something obvious.