- #1

Leb

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## 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...