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abstract algebra help

  • 15-04-2008 10:33am
    #1
    VaccinalBowl
    Closed Accounts Posts: 20 ✭


    So what I want to do is know how to determine the invertible elements of a monoid. I'll give you guys my question so you can see what I gotta do but it's only the last part I'm stuck on.

    Let ~ be the binary operation on the set RXR of ordered pairs of real numbers definded such that (a,b) ~ (c,d) = (ac, d +bc) for all odered pairs (a,b) and (c,d) of real numbers. Prove that (RxR, ~) is a monoid. I've done this part. What is the identity element of this monoid?Work this out to be (1,0) Which are the inverible elements of this monoid?This is the part I'm stuck on. I don't know how to decide what the invertible elements of the group are. and I need to be able to do this to get the answer to the next part!


Comments

  • #2
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭Fremen


    An element (a,b) will be invertible if there exists some (c,d) such that

    (a,b) ~ (c,d) = (1, 0)
    (c,d) ~ (a,b) = (1, 0)

    This gives

    ac = ca = 1
    d + bc = 0
    b + ad = 0

    Seems like the only restriction in picking numbers which satisfy the above is that a and c are nonzero.


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