Varieties of Automorphism Groups of Orders
W. Charles Holland · Transactions of the American Mathematical Society · 1985
The group $A(\Omega )$ of automorphisms of a totally ordered set $\Omega$ must generate either the variety of all groups or the solvable variety of class $n$. In the former case, $A(\Omega )$ contains a free group of rank ${2^{{\aleph _0}}}$; in the latter case, $A(\Omega )$ contains a free solvable group of class $n - 1$ and rank ${2^{{\aleph _0}}}$.