Let $G$ be a group acting on a set $\Omega$ faithfully. Then 2-closure of $G$ denoted by
$G^{(2)}$ is the largest subgroup of the symmetric group of $\Omega$ with
the same orbits as $G$ on $\Omega\times\Omega$. Clearly $G\leq G^{(2)}$. If $G=G^{(2)}$ then $G$ is called 2-closed. I am collecting some group-theoretic properties of $G$ which inherited to $G^{(2)}$. For example, I know that
if $G$ is abelian (p-group,nilpotent, or has odd order), then so is $G^{(2)}$.
Also I am looking for (group-theoretic) properties of $G$ which makes it to be 2-closed.
For example, I know that every semiregular permutation group is 2-closed.
Also I guess that every cyclic permutation group is 2-closed.
I know that the basic properties of closures of permutation groups
are given in Wielandt's book, "Permutation groups through invariant relations and invariant
functions, lectures given at The Ohio State University, Columbus, Ohio,
1969", but unfortunately I dont have access it. Now my questions are requests are:
1) Which group-theoretic properties of $G$ inherited to $G^{(2)}$?(other than
the above properties) For example, for which solvable groups $G$, $G^{(2)}$
is solvable?
2) Which group theoretic properties of $G$ make it 2-closed?(other than when
$G$ is cyclic)
3) may it possible that one send me the file of Wielandt's book?



2-closure of a permutation group
Let's call an abstract group $2$-closed if all of its faithful permutation representations are $2$-closed.


You observed that cyclic groups are $2$-closed.


Also a group $G$ for which the only faithful permutation representation is the regular representation must be $2$-closed, because in any faithful action there must be a regular orbit, so if $x$ is in this orbit then the stabilizer of $x$ in $G^{(2)}$ must also be trivial, so $G = G^{(2)}$.


The only finite groups with that property are cyclic groups of prime power order and (generalized) quaternion groups. (I am not sure about infinite examples, but the group ${\mathbb Z}\_{p^\infty}$ is one such for each prime $p$.)


So the (generalized) quaternion groups $Q\_{2^n}$ for $n \ge 3$ are $2$-closed and I think the direct product of $Q\_{2^n}$ with any cyclic group of odd order is also $2$-closed.


I have not been able to think of any other finite examples. $Q\_{2^n} \times C\_2$ is not $2$-closed for example. But I would not care to make a conjecture about it, even for finite $p$-groups.


