
> 
> Consider the additive abelian group $(\mathbb{R},+)$. Does there exists a binary operation $\circ:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that the following holds
> 
> 
> * $(\Bbb{R},\circ)$ is a group.
> * For all $S\subseteq \mathbb{R}$ the subgroup generated by $S$ in $(\mathbb{R},+)$ is **equal** (as a set) to the subgroup generated by $S$ in $(\mathbb{R},\circ)$.
> * $(\mathbb{R},+)$ is not isomorphic to $(\mathbb{R},\circ)$.
> 
> 
> 


This question is a follow up of [this MSE post](https://math.stackexchange.com/questions/2892167/isomorphism-of-two-groups-on-the-same-set-via-the-knowledge-of-subgroups) of mine and due to it I guess that the answer to both the questions are negative. But I can't construct such a group (or prove the existence of such one).


Thanks for any help.



Does there exists a group structure on $\circ$ on $(\mathbb{R},\circ)$ such that $(\mathbb{R},\circ)$ is non-isomorphic to $(\mathbb{R},+)$?
The answer is no: every such group law is isomorphic to the standard law.


Let me prove something stronger: the poset structure of the lattice of subgroups $\mathrm{Sub}(\mathbf{R})$ characterizes $\mathbf{R}$ up to group isomorphism.


Let us first check it among abelian groups. Let $G$ be a group with $\mathrm{Sub}(G)\simeq\mathrm{Sub}(\mathbf{R})$ as posets. The absence of nontrivial minimal elements characterizes torsion-freeness, so $G$ is torsion-free. Since $G$ is abelian, the absence of nontrivial maximal element characterizes the absence of proper subgroups of finite index, and hence $G$ is divisible. So $G$ is isomorphic to $\mathbf{Q}^{(I)}$ for some cardinal $I$. Cyclic subgroups can be characterized, among abelian groups, as those groups with infinitely many subgroups, and in which every nontrivial subgroup is contained in only finitely many subgroups. The number of cyclic subgroups of $\mathbf{Q}^{(I)}$ for nonempty $I$ is equal to $\max(\aleph\_0,|I|)$. So here we have $|I|=c$. 


This concludes the abelian case.


In the nonabelian case, we need two lemmas (of independent interest, and probably proved somewhere in the literature).


(1) the poset structure of the lattice of subgroups $\mathrm{Sub}(\mathbf{Z}^2)$ characterizes $\mathbf{Z}^2$ up to group isomorphism.


In a lattice $X$ with maximal element $q$, let us say that $x$ satisfies Property $A\_p$ ($p$ a given prime) if the interval $[x,q]$ consists of $x$, $q$, and exactly $p+1$ pairwise non-comparable elements. Then in $\mathrm{Sub}(\mathbf{Z}^2)$, it is easy to check that the only subgroup satisfying $A\_p$ is $p\mathbf{Z}^2$.


Consider $G$ with $\mathrm{Sub}(G)\simeq\mathrm{Sub}(\mathbf{Z}^2)$. Then $G$ has a unique subgroup $G\_p$ satisfying $A\_p$ for every $p$, and $\bigcap\_p G\_p=\{1\}$ (since the corresponding property holds in $\mathbf{Z}^2$). In particular, $G\_p$ is a normal subgroup. The quotient $G/G\_p$ has finitely many subgroups and hence is a finite group. So $G$ is residually finite.


If by contradiction $G$ were non-abelian, there would exist $p$ such that $G/G\_p$ is non-abelian; take $p$ minimal for this.
The finite group $G/G\_p$ is nonabelian and has all its proper subgroups cyclic of prime order. The only possible groups having this property have the form $C\_q\ltimes C\_{q'}$ with $q,q'$ prime and $q$ divides $q'-1$. It has $q'+3$ subgroups ($q'+1$ is we exclude the trivial and whole subgroups), so $q'=p$. Now $G/(G\_p\cap G\_q)$ inherits from the corresponding group in $\mathbf{Z}^2$, the property of having exactly $(p+3)(q+3)$ subgroups, including $p+q+2$ maximal ones. On the other hand, in $G$, the uniqueness of a subgroup with Property $A\_q$ implies that $C\_q^3$ is not a quotient of $G$; it follows that $G/(G\_p\cap G\_q)$ has the form $C\_q^2\ltimes C\_p$. This group has $q+1$ maximal subgroups that are normal, and $p$ maximal subgroups that are non-normal (the $q$-Sylow subgroups), so the total is $p+q+1$, which does not match: contradiction.


Now $G$ is abelian, torsion-free. The (at most) uniqueness of subgroups with Property $A\_p$ therefore applies to subgroups of $G$ and hence $G$ has $\mathbf{Q}$-rank $\le 2$. Since $G/G\_p\simeq C\_p^2$ for all $p$, we deduce $G\simeq\mathbf{Z}^2$. $\Box$


(2) the poset structure of the lattice of subgroups $\mathrm{Sub}(\mathbf{Z})$ characterizes $\mathbf{Z}$ up to group isomorphism.


Let $G$ be such a group. Then $G$ is torsion-free. Say that a subgroup has Property B if it is contained in finitely many subgroups. From the property in $\mathbf{Z}$, we deduce that every nontrivial subgroup has Property B, and that the intersection of any two nontrivial subgroups has Property B. We deduce that $G$ has a commensurated subgroup $N$ isomorphic to $\mathbf{Z}$. The germ at infinity of the action on $N$ yields a homomorphism $G\to\mathbf{Q}^\times$. The image has to be contained in $\{\pm 1\}$: otherwise some element of $G$ acts by some element $\neq\pm 1$, and hence generates a cyclic subgroup with trivial intersection with $N$. Also, since $N$ has Property B, we see that $G$ is finitely generated. We deduce that some finite index subgroup of $N$ is normal, say $N$ itself (up to rename). Since $G/N$ has finitely many subgroups, it is finite, and hence $G$ is virtually cyclic and torsion-free (and infinite). This is known to imply that $G$ is infinite cyclic. $\Box$


Now let us conclude. Consider $G$ with $\mathrm{Sub}(G)\simeq\mathrm{Sub}(\mathbf{R})$. So $G$ is torsion-free. By the lemma, we can recognize subgroups of $G$ isomorphic to $\mathbf{Z}$ and $\mathbf{Z}^2$; in particular, we inherit from $\mathbf{R}$ the property that any two infinite cyclic subgroups is contained in a subgroup isomorphic to $\mathbf{Z}^2$. Hence $G$ is abelian. Since the abelian case is done, this ends the proof.




---


Edit: Here's a reference: the book by M. Suzuki, *Structure of a Group and the Structure of its Lattice of Subgroups, Ergebnisse der Mathematik und Ihrer Grenzgebiete 10, Springer-Verlag Berlin Heidelberg, 1956*


Theorem 3, page 35 (attributed to R. Baer, 1939), says that for any abelian group $G$ of $\mathbf{Q}$-rank $\ge 2$ (i.e., containing a subgroup isomorphic to $\mathbf{Z}^2$), any "projectivity" of $G$ (that is, an isomorphism $\mathrm{Sub}(G)\to\mathrm{Sub}(H)$ for another group $H$), is induced by a unique (up to pre-composition by $x\mapsto -x$) isomorphism $G\to H$. In particular, $H$ is isomorphic to $G$.


(While the "in particular" assertion holds for $G=\mathbf{Z}$ by Lemma (2) above, the stronger assertion is false for $G=H=\mathbf{Z}$, as the lattice of subgroups has an uncountable automorphism group then.)


