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

[Gear300]

What kind of operation exists that is valid for a group limited to the natural numbers?

 

[VKint]

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

$$\displaystyle a \star b \equiv <br /> \left\{<br /> \begin{aligned}<br /> 2f(a,b) \quad \textrm{if} \; f(a,b) \geq 0\\<br /> |2f(a,b)| + 1 \quad \textrm{if} \; f(a,b) < 0 \textrm{.}<br /> \end{aligned}<br /> \right.$$

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

The natural structure of $$\mathbb{N}$$ 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 $$\mathbb{N}$$ as a group in some way?

 

[Gear300]

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

$$\displaystyle a \star b \equiv <br /> \left\{<br /> \begin{aligned}<br /> 2f(a,b) \quad \textrm{if} \; f(a,b) \geq 0\\<br /> |2f(a,b)| + 1 \quad \textrm{if} \; f(a,b) < 0 \textrm{.}<br /> \end{aligned}<br /> \right.$$

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

The natural structure of $$\mathbb{N}$$ 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 $$\mathbb{N}$$ as a group in some way?

Thanks for the help.
The reason I asked was simply out of curiosity (while thinking, I couldn't think of a naturally occurring one).