Question

Let H be a subgroup of a group G. Let ∼H and ρH be the equivalence...

Let H be a subgroup of a group G. Let ∼H and ρH be the equivalence relation in G introduced in class given by

x∼H y⇐⇒x−1y∈H, xρHy⇐⇒xy−1 ∈H.
The equivalence classes are the left and the right cosets of H in G, respectively. Prove that the functionφ: G/∼H →G/ρH given by
φ(xH) = Hx−1

is well-defined and bijective. This proves that the number of left and right cosets are equal.

Math   Advanced-Math   
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Answer #1

Well-defined:

Where

Consider

So that

But

So that

And so

Meaning is well defined

For bijectivity: let

Then so

Therefore the given function is both well defined and bijective

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!

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