Theorem about cosets - what am I doing wrong

[murmillo]

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?

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.

 

[Tedjn]

Here's a case when notation helps. The law of composition is addition, so cosets are actually 4+H and 12+H.

 

[murmillo]

Right, but now I'm confused as to something I've read before:

"Let us denote the subset of Z consisting of all multiples of a given integer b by bZ:
bZ = {n in Z | n = bk for some k in Z}."

When the author writes n = bk, he does not mean b+k, even when the law of composition is addition. In this case bk = b+b+b+b+... (k terms).

This is why I thought that in my example, even if the law of composition of Z is additive, by 4H is meant {n in H } | n = 4h for some h in H} where 4h means h+h+h+h, i.e. 4 times h, not 4+h.

So I'm still confused.

Oh, Artin is so not the best book for self-studying algebra...

 

[hunt_mat]

There is an equivalence relation a\sim b <=>a\in bH equivalence relations form partitions. there are other details but you will have fun writing the words down.

Mat

 

[murmillo]

hunt mat: I think I'm having more trouble with the notation mentioned in the last post. Once I understand the confusing notation, the proof should be easy. But I'm still having trouble with notation.

 

[vela]

You can't change the subgroup midstream. Say you have the subgroups H₁=5Z and H₂=10Z. The proposition says that if H₁+m and H₁+n have an element in common, they are equal. Similarly, if H₂+m and H₂+n have an element in common, they are equal. But the proposition doesn't say anything about how the cosets of H₁ and the cosets of H₂ are related even if they have elements in common.

 

[vela]

Say H=5Z. Then you have H={..., -10, -5, 0, 5, 10, ...} and

H+1 = {..., -9, -4, 1, 6, 11,...}
H+2 = {..., -8, -3, 2, 7, 12,...}
H+3 = {..., -7, -2, 3, 8, 13,...}
H+4 = {..., -6, -1, 4, 9, 14,...}
H+5 = {..., -5, 0, 5, 10, 15,...} = H

Does that make sense?

 

[murmillo]

What you're saying makes sense. I think the only thing I don't understand is why addition is used instead of multiplication. Let me explain.

In my original post I was not trying to refer to 2 different subgroups. I was referring to only one, H = 5Z. I think my confusion is in this definition:

"A left coset is a subset of the form aH = {ah | h is in H}."

You claim that if H is 5Z, then by aH is meant H+a = {a+h | h is in H}. Since the law of composition on Z is addition. Correct?

But I'm confused because in previous sections he wrote:

"Let us denote the subset of Z consisting of all multiples of a given integer b by bZ:
bZ = {n in Z | n = bk for some k in Z}."

When the author writes n = bk, he does not mean b+k, even when the law of composition is addition. In this case bk = b+b+b+b+... (k terms).

Why does bk mean b+b+b+...(k terms) instead of b+k in this previous section? Why does it mean b+k when referring to cosets?

 

[hunt_mat]

Getting an abstract answer is sometimes easier to digest.

 

[boboYO]

Because when talking about abstract groups "ab" always means "the law of composition applied to the pair of elements (a,b) ".

When working in Z he uses the normal notation for multiplication because you're used to it.

 

[vela]

What you're saying makes sense. I think the only thing I don't understand is why addition is used instead of multiplication. Let me explain.

In my original post I was not trying to refer to 2 different subgroups. I was referring to only one, H = 5Z. I think my confusion is in this definition:

"A left coset is a subset of the form aH = {ah | h is in H}."

You claim that if H is 5Z, then by aH is meant H+a = {a+h | h is in H}. Since the law of composition on Z is addition. Correct?

But I'm confused because in previous sections he wrote:

"Let us denote the subset of Z consisting of all multiples of a given integer b by bZ:
bZ = {n in Z | n = bk for some k in Z}."

When the author writes n = bk, he does not mean b+k, even when the law of composition is addition. In this case bk = b+b+b+b+... (k terms).

Why does bk mean b+b+b+...(k terms) instead of b+k in this previous section? Why does it mean b+k when referring to cosets?

It's just an unfortunate overloading of notation. Though bZ and aH appear similar notation-wise, they mean different things. The definition of bZ has nothing to do with what aH means.

 

[murmillo]

OK, I think I understand. I was mixing up two different notations. By bZ is mean the subset of Z consisting of all multiples of a given integer b by bZ:
bZ = {n in Z s.t. n=bk for some k in Z} and bk meant do b+b+b... k times. That's completely different by what was meant by aH = {ah s.t. h is in H}, since by ah is meant a*h where * is the law of composition on G. Right?

 

[vela]

Right.

 

[murmillo]

Oh wait. But he also says, "The subgroup H is itself a coset, because H = 1H." By that he doesn't mean 1+H. What's the explanation?

edit: never mind. Z is completely out of the picture.

By 1 he meant the identity! Oh goodness I should've used another textbook.

Now I can finally go on! Thanks, everyone!