If G is a group with subgroups A and B, then how do you define "x ~y" if and only if there exists a in A and b in B such that x = ayb?
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that IS the definition. i suspect what you want to do is show "~" is an equivalence relation; that is, show ~ is reflexive, symmetric and transitive.
ok so R is an equivalence relation Its reflexive: y ~ y Symmetry: x ~ y then y ~ x is it like this? i don't get the part "if and only if there exists a in A and b in B such that x = ayb" what do i do with x = ayb
to prove that ~ is reflexive, you need to prove that for ANY x in G, you can find a in A and b in B such that x = axb. you can't just say that x~x, because you don't KNOW that, unless you can actually produce the elements a and b.
ok got it. Thank you
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