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An example of a nontrivial group $G$ such that $2^n$ divides $|G|$ for all positive integers $n$
It is well known that if $G$ is a torsion free Abelian group, then $2^n$ divides $|G|$ for all positive integers $n$.
In fact, in his 1955 paper, G. W. Leech proved the stronger result that if $G$ is torsion free Abelian of finite rank, then $|G|=2^m$ for some positive integer $m$, and if $A$ and $B$ are two infinite discrete torsion free Abelian groups, then $|A\oplus B|=2^m$, where $m$ is the maximum of the orders of finite subgroups of $A$ and $B$.
In the non-Abelian case, there is no apparent general upper bound for the order of a finitely generated torsion free group, so that it is unlikely that there exists an example with the property that $2^n$ divides the order of the group for every positive integer $n$.
Is there an example of a torsion free group $G$ such that $2^n$ divides the order of $G$ for every positive integer $n$?
A:
A result of Swan gives a decomposition of any group of exponent 2. (Actually, the exponent does not need to be $2$, but results exist only for exponents in this range.) Specifically, it is known that $G$ admits a regular decomposition, and if $G=G_1\rtimes G_2\cdots \rtimes G_r$ is any regular decomposition, then the exponent of $G$ is the maximum exponent of the $G_i$'s. This implies that if the largest
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