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Subgroup proof

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Let A and B be finite subgroups of G.
Even though AB need not be a subgroup of G, show that
|AB||A n B| = |A||B|.
(Hint: Define (a_1,b_1)~(a_2,b_2) iff a_1b_1 = a_2b_2.
Prove that ~ is an equivalence relation and examine the equivalence
classes.)

I need a detailed and rigorous proof to study for a test please.

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Solution Summary

This provides an example of completing a subgroup proof.

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please see the attached file

Let us define as suggested a binary relation on .
This is an equivalence relation: it is clearly reflexive, symmetric and transitive.
Therefore can be partitioned into equivalence ...

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