Proof: Cosets equal or disjoint

In summary: So, since aH = bH, we can say that b is in aH, and then b = ah for some h ∈ H. This means that b ∈ aH, and also that b = ah for some h ∈ H.In summary, Two left cosets aH, bH of H in G are equal if and only if a^{−1}b ∈ H and b ∈ aH. This can be proven by assuming aH = bH and showing that this implies b ∈ aH and b = ah for some h ∈ H.
  • #1
Leb
94
0

Homework Statement



Two left cosets aH, bH of H in G are equal if and only
if a^{−1}b ∈ H. This is also equivalent to the statement b ∈ aH.
Proof:
Suppose that aH = bH. Then e ∈ H. So, b = be ∈ bH. If
aH = bH then b ∈ aH. So, b = ah for some h ∈ H. But, solving for
h, we get h = a
−1
b ∈ H


Homework Equations


Let G be a group, H a subgroup. a,b ∈ G.
Def: Left coset aH={ah |h ∈ H}

Attempt:
I never managed to understand this simple concept of cosets, but maybe understanding this will help. I do not see why when saying that aH = bH implies that b ∈ aH, which implies b = ah for some h ∈ H. That is how {ah |h ∈ H} = {bh |h ∈ H} implies b ∈ aH and b=ah, and not bh ∈ aH and bh =ah ? The definition of the coset only gives information on h, not on b...
 
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  • #2
Leb said:

Homework Statement



Two left cosets aH, bH of H in G are equal if and only
if a^{−1}b ∈ H. This is also equivalent to the statement b ∈ aH.
Proof:
Suppose that aH = bH. Then e ∈ H. So, b = be ∈ bH. If
aH = bH then b ∈ aH. So, b = ah for some h ∈ H. But, solving for
h, we get h = a
−1
b ∈ H

Homework Equations


Let G be a group, H a subgroup. a,b ∈ G.
Def: Left coset aH={ah |h ∈ H}

Attempt:
I never managed to understand this simple concept of cosets, but maybe understanding this will help. I do not see why when saying that aH = bH implies that b ∈ aH, which implies b = ah for some h ∈ H.
It is the same as you have in the first proof above. H is a subgroup so the identity, e, is in H and therefore bH contains b. Since we are given that aH= bH, b is in aH. Any member of aH is of the form ah so we have b= ah for some h in H.

That is how {ah |h ∈ H} = {bh |h ∈ H} implies b ∈ aH and b=ah, and not bh ∈aH[/quote]
What h are you talking about? Certainly it is true that bh∈ aH for some h in H- specifically h= e.

and bh =ah ?
If that were true then we could multiply both sides of the equation, on the right, by h-1 and get b= a. Saying that bh= a or b= ah', possibly with h= h' or [itex]h\ne h'[/itex], does not lead to ah= bh. Any member of

The definition of the coset only gives information on h, not on b...
Yes, of course, b is "given" and does not change. You don't need to know any thing more about b. The only information on h you need (or have) is that it is a member of H.
 
  • #3
Thanks for the reply!
I think I figured it out just after I posted the message, I think my problem was that I missed the "Suppose that aH = bH" part, which was crucial for the proof to work.
 

Related to Proof: Cosets equal or disjoint

1. What is a coset in group theory?

A coset in group theory is a subset of a group that is obtained by multiplying every element in a subgroup by a fixed element in the group. It can also be thought of as a shift or translation of the subgroup.

2. How do you prove that two cosets are equal?

To prove that two cosets are equal, you must show that every element in one coset is also in the other coset, and vice versa. This can be done by showing that the two cosets have the same cardinality and that they share at least one common element.

3. Can two cosets be disjoint?

Yes, two cosets can be disjoint if their corresponding subgroups do not share any elements. This means that the two cosets would have no elements in common and therefore, would be disjoint.

4. What is the significance of proving that two cosets are equal or disjoint?

Proving that two cosets are equal or disjoint allows us to better understand the structure of a group. It helps us identify the relationships between different subgroups and how they interact with each other within the larger group.

5. How does proving that cosets are equal or disjoint relate to the Lagrange's Theorem?

Proving that cosets are equal or disjoint is a crucial step in the proof of Lagrange's Theorem, which states that the order of a subgroup must divide the order of the larger group. This theorem helps us understand the structure and properties of groups, and cosets play a key role in its proof.

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