Can one make a group out of these transcendental points?

Question

Can one make a group out of a set of transcendental points including the identity point? Here, $\Bbb A$ represents any non-zero algebraic number. I'm working with real numbers.

My set consists of points $X=\{(e^{\Bbb {-A}},e^{\Bbb {-A}}), (e^{\Bbb {A}}, e^{\Bbb A}),(1,1)\}.$ I have the identity element as $(1,1)$ and I've checked that associativity holds and commutativity holds. I have the binary operation as multiplication.

By Lindemann-Weirstrauss, $e^\alpha$ is transcendental whenever $\alpha$ is non-zero algebraic.

To combine elements I take the Cartesian product of the first two points in the set defined above. That is, $(e^{\Bbb {-A}},e^{\Bbb {-A}})\times(e^{\Bbb {A}}, e^{\Bbb A}).$

Answer 1

A group is a set $G$ together with a binary operation, a map $G\times G\to G$, satisfying certain axioms. In your case, you have a 3-element set $G$ (which you call $X$). Your binary operation $*$ is the coordinate-wise multiplication. In order for your operation to be a map $G\times G\to G$, you have to have: $$ (e^A, e^A) * (e^A, e^A) \in G\Rightarrow e^A\cdot e^A\in \{1, e^A, e^{-A}\}. $$ Thus, either $e^{2A}=1$, or $e^{2A}=e^A$, or $e^{2A}=e^{-A}$. In each of these cases, $A=0$, i.e. $e^A=1$. In other words, the only time your $G$ with the given binary operation is a group is when $G=\{(1,1)\}$ and $e^A=1$, i.e. is not a transcendental number.

Few more things:

1. The last paragraph in your question is very unclear since in a group there is no "combination" operation.

2. Your question "Can one make a group out of a set of transcendental points including the identity point?" has the affirmative answer: Take any transcendental number $\alpha$ (real or complex, does not matter) and consider the subgroup $\langle \alpha\rangle=\{\alpha^n: n\in {\mathbb Z}\}$ in ${\mathbb C}^\times$ generated by $\alpha$. Then $\langle \alpha\rangle$ with the binary operation given by the usual multiplication is a group, consisting only of transcendental numbers (and $1$).

3. On a side note, this algebra question is better than the topology questions you asked earlier. My suggestion is to pick up an algebra textbook with exercises (see here for suggestions) and solve them one-by-one. Then do the same with a topology textbook, say, Munkres.