A group $G$ is called a $k$-orbit group if its automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as a (nonzero) vector space $V$: clearly $\operatorname{GL}(V)\leqslant\operatorname{Aut}(V)$ has 2 orbits on $V$. By "blowing up the field" $V$ may be viewed as a vector space over the prime field (see comments by Rickard and YCor).

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as a (nonzero) vector space $V$ over a field $F$: clearly $GL(V/F)\leqslant{\rm Aut}(V)$ has 2 orbits $\{0\}$ and $V\setminus\{0\}$ as Rickard noted. Viewing $V$ as a vector space over the prime field of $F$ gives YCor's comment.

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as a (nonzero) vector space $V$: clearly $GL(V)\leqslant{\rm Aut}(V)$ has 2 orbits on $V$. By "blowing up the field" $V$ may be viewed as a vector space over the prime field (see comments by Rickard and YCor).

A group $G$ is called a $k$-orbit group if its automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as a (nonzero) vector space $V$: clearly $\operatorname{GL}(V)\leqslant\operatorname{Aut}(V)$ has 2 orbits on $V$. By "blowing up the field" $V$ may be viewed as a vector space over the prime field (see comments by Rickard and YCor).

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as (nonzero) vector space $V$: clearly $GL(V)\leqslant{\rm Aut}(V)$ has 2 orbits on $V$. By "blowing up the field" $V$ may be viewed as a vector space over the prime field (see comments by Rickard and YCor).

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_p)^n$ where $p$ is prime and $n\geqslant1$.

Question 1: Is it hopeless to classify the infinite 2-orbit groups?

The automorphism group of the additive group $D^+$ of a division algebra $D$ is transitive on the set $D^\times=D\setminus\{0\}$: if $a,b\in D^\times$, then $a^\alpha=b$ where $\alpha\in{\rm Aut}(D^+)$ sayisfies $x^\alpha=xa^{-1}b$. (Taking $D=\mathbb{F}_{p^n}$ gives the elementary abelian example.) However, Corollary 2 on page 212 of "P.M. Cohn, The embedding of firs in skew fields. Proc. London Math. Soc. (3) 23 (1971), 193–213" shows that there is a division algebra $D$ such that the multiplicative group $D^\times$ has 2 conjugacy classes (1 non-trivial). Hence $D^\times$ is a non-abelian 2-orbit group. More generally, HNN extensions show that there exist infinite groups $G$ which I would guess do not equal $D^\times$ for any $D$, with precisely 2 conjugacy classes, see "G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247–254". (Cohn's construction mimics HNN's.) I would suppose that there are many other rather different examples of 2-orbit groups; however, I am unaware of different constructions.

Edit: The abelian example is better framed as a (nonzero) vector space $V$: clearly $GL(V)\leqslant{\rm Aut}(V)$ has 2 orbits on $V$. By "blowing up the field" $V$ may be viewed as a vector space over the prime field (see comments by Rickard and YCor).