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If A and B are subgroups of G and AB=G does it follow that AB=BA? If yes, why?
What does AB mean, i mean what operation are you performing between A and B here? is this supposed to be the same operation of the group G, or?If A and B are subgroups of G and AB=G does it follow that AB=BA? If yes, why?
AB is the set of elements of the form ab (with a in A & b in B).What does AB mean, i mean what operation are you performing between A and B here? is this supposed to be the same operation of the group G, or?
it doesn't mean I can see it.tgt, can you give an example in which AB= G and BA= {1}?
Your proof should have stopped here - this is all you need to show. AB=BA is an equality of sets.ok, so what about this.
Like said AB is the set of all elements such a is in A and b is in B. Then from this follows that also [tex] a^{-1}b^{-1}\in AB, since, a^{-1}\in A, b^{-1}\in B[/tex] this comes from the fact that both A,B are subgroups , so they do have inverses.
NOw since AB=G, it means also that the inverse of [tex] a^{-1}b^{-1}[/tex] is in AB. that is :
[tex] (a^{-1}b^{-1})^{-1}=ba \in AB[/tex]
This step is wrong. Can you see why?[tex]b^{-1}(ba)=a[/tex] now multiplying with b we get
[tex] ba=ab[/tex]
Hell YES! This happens when you assume that the same thing is gonna happen in next step as well. We cannot do this, because we dont know whether A, B are commutative. so by doing what i did in the last step, i should have multiplied by b from the left side, which would get us nowhere, since we would end up with ba=ba... and this is not what we wanted.This step is wrong. Can you see why?