Is $ \lbrace e^{\large\frac{s}{\log x}} : s \in \Bbb R \rbrace $ a useless group?

Question

Given the group: $ G_1 = \lbrace e^{\large\frac{s}{\log x}} : s \in \Bbb R \rbrace $, with multiplication as the operation and $1$ as the identity element,

Is this group of any interest? It makes a correspondence between elements of the form: $ e^{\large\frac{s}{\log x}} $ and real numbers $s$, but couldn't you just use the real numbers as a group just as easily? Isn't it just an overly complicated group?

Edit:

How does $ G_2 $ compare to $ G_1 $, where $G_2 = \lbrace \zeta(s)^{\large\frac{1}{\log x}} : s \in \Bbb R \rbrace $, where $\zeta(s) $ is the Riemann zeta function. There is only one inverse for all $s$ correct?

Answer 1

Well, inherently the group $G_1$ you are describing is just the positive real numbers under multiplication. You just have a rather complicated way of describing it.

However, I think it's worth pointing out that, beyond the literal meaning of the symbols, there is something of substance here: Implicitly, you are thinking of a map $\mathbb R\rightarrow \mathbb R^+$ given by $$s\mapsto x^s$$ where $x$ is a positive real number. This is a group homomorphism (and, in fact, an isomorphism for $x\neq 1$). This map is an isomorphism, which means that, as far as group structure goes, the positive real numbers under multiplication act the same as real numbers under addition. So, from some points of view, yes, this is just an odd way of referring to the usual group $\mathbb R$.

A common theme in mathematics is that isomorphic objects can carry different implications - note that there are many isomorphisms between $\mathbb R$ and $\mathbb R^+$, so they can't be canonically identified. For instance, if you were doing geometry and wanted to consider, say, the set of dilations from a point as a group, you would naturally end up working in $\mathbb R^+$ rather than $\mathbb R$. So, I wouldn't say that it is useless, even if it's isomorphic to a more common group.

Answer 2

This should really be a comment, but it's too long.

The group (for $x$ positive and $\not=1$) is just $(\mathbb{R}_{>0},\times)$, described in an unnecessarily messy way. However, there is a neat map involved here.

For every positive $x$ other than $1$, the map $$\varphi_x: s\mapsto e^{s\over\log x}$$ is a group isomorphism from $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$. So in a certain sense, what your notation is really doing is describing a continuously$^1$ varying group: to each positive $x$ other than $1$, you've assigned a copy $C_x$ of $(\mathbb{R}_{>0},\times)$, and they're intuitively "stretching" as we vary $x$. For example, $\varphi_e(1)=e$ and $\varphi_{e^2}(1)=\sqrt{e}$, so in some sense the $e$ of $C_e$ corresopnds to the $\sqrt{e}$ of $C_{e^2}$. Indeed, we can picture a "thread" running through the groups $C_x$ corresponding to the number $1$ in $(\mathbb{R},+)$.

This sort of "continuously varying object" is actually quite important in mathematics. Elsewhere you'll see the same basic idea (broadly speaking) occurring in many contexts. As an example, you might think about how to view a Mobius strip as a "continuously (over a circle) varying line" - that is, a line bundle over a circle.

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$^1$You may not be familiar at this point with the subject of topology, but the point is that all the pieces involved in this picture have a sort of "geometry" associated to them, and all the transformations we're describing here are "continuous" in the sense of that geometry. It's not important at this stage to dive into that if you're not already familiar with the basics; for now, just see if you can get an intuitive sense that when we change $x$ very slightly, the copy $C_x$ only distorts "a little" (somewhat more concretely, if $x$ is very close to $y$ then $e^{s\over \log x}$ is very close to $e^{s\over\log y}$).