1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Operation exists that is valid for a group limited to the natural

  1. Jul 16, 2009 #1
    What kind of operation exists that is valid for a group limited to the natural numbers?
  2. jcsd
  3. Jul 16, 2009 #2
    Re: Groups/Operations

    I can't think of any really natural or appealing ones, but you can easily cook up some fairly contrived group operations on [tex] \mathbb{N} [/tex] by bijecting it with [tex] \mathbb{Z} [/tex]. For example, for [tex] a,b \in \mathbb{N} [/tex], define the function [tex] f(a,b) [/tex] as follows:

    \displaystyle f(a,b) \equiv (-1)^{2 \left\{ \frac{a}{2} \right\} } \left\lfloor \frac{a}{2} \right\rfloor + (-1)^{2 \left\{ \frac{b}{2} \right\} } \left\lfloor \frac{b}{2} \right\rfloor \textrm{.}

    Then let

    \displaystyle a \star b \equiv
    2f(a,b) \quad \textrm{if} \; f(a,b) \geq 0\\
    |2f(a,b)| + 1 \quad \textrm{if} \; f(a,b) < 0 \textrm{.}

    Then [tex] \star [/tex] is a group operation, with identity element [tex] 1 [/tex], such that [tex] n^{-1} = n + (-1)^n [/tex].

    The natural structure of [tex] \mathbb{N} [/tex] is that of a monoid, i.e., a "group without inverses." By the way, just out of curiosity, why do you ask? Is there some problem you're working on that requires you to treat [tex] \mathbb{N} [/tex] as a group in some way?
  4. Jul 17, 2009 #3
    Re: Groups/Operations

    Thanks for the help.
    The reason I asked was simply out of curiosity (while thinking, I couldn't think of a naturally occurring one).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook