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Math Help - define x ~ y

  1. #1
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    define x ~ y

    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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  2. #2
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    Re: define x ~ y

    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.
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  3. #3
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    Re: define x ~ y

    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
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  4. #4
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    Re: define x ~ y

    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.
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  5. #5
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    Re: define x ~ y

    ok got it. Thank you
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