Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Groups do not necessarily have to have only one operation

  1. Mar 28, 2009 #1
    To be sure of things, groups do not necessarily have to have only one operation; they may have more, but there is one and only one operation they are identified with. Am I right?
  2. jcsd
  3. Mar 28, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Re: Groups

    There may be two operations o and * defined on a set G, such that (G, o) and (G, *) are both groups.
    When we say "let G be a group" we're sloppy, you should say what the operation is.
  4. Mar 28, 2009 #3
    Re: Groups

    Thus, if I were asked whether the set of all rational numbers Q with addition was a group, would it be valid to assume multiplication as an operation and state that a/b + c/d = (ad + cb)/(bd) ε Q (for a, b, c, d ε Z) in the case of proving that addition is closed with respect to Q.
  5. Mar 28, 2009 #4


    User Avatar
    Science Advisor
    Homework Helper

    Re: Groups

    Yes, that's valid, but you're not multiplying group elements (which would be bad). You're just expanding the definition of rational addition.
  6. Mar 28, 2009 #5
    Re: Groups

    I see. Thanks for the replies.
  7. Mar 28, 2009 #6
    Re: Groups

    Well, in general there is only one operation that is defined in a group. That is there is only one operation which is applied berween the elements of a group.

    In your example, we could say let (Q,@) be a group, where Q is the set of rational numbers, and then we would say that @ is defined in this way:

    [tex]\frac{a}{b} @ \frac{c}{d}=\frac{a*d+c*b}{b*d}[/tex]

    where + is the natural addition symbol and * multiplication.

    But, like it was said above, here you are not multiplying the elements of Q, which we have assumed are of the form

    [tex]Q={ \frac{a}{b}: a,b \in Z }[/tex]
  8. Mar 29, 2009 #7


    User Avatar
    Science Advisor

    Re: Groups

    But in any case, in
    [tex]\frac{a}{b}+ \frac{c}{d}= \frac{ad+ bc}{cd}[/tex]
    You are NOT multiplying rational number you are multiplying integers.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Groups necessarily operation Date
I What is the symbol for "not necessarily imply" Mar 11, 2018
B Index numbers vs. Quantity in a group Oct 13, 2017
Insights Groups and Geometry - Comments Jun 30, 2016
I Help with proofs in general May 12, 2016
I Group theory Apr 18, 2016