Comment #18 on issue 2039 by asmeurer: Mul.eval_subs problems http://code.google.com/p/sympy/issues/detail?id=2039
Here's a simple example. The smallest non-abelian (non-commutative) group is S3, the group of permutations on 3 elements. You can see from the multiplication table at (http://en.wikipedia.org/wiki/Dihedral_group_of_order_6#Summary_of_group_operations), looking down the main diagonal, f is the only element that has a square root. Every other element when squared gives the identity element (e), which also implies that sqrt(e) cannot be well-defined in this group. It's easy to come up with an example of an infinite group that has no square-roots, just take Q, the set of rational numbers. Then the element 2 will not have a square-root in Q. I am sure that there are examples of infinite non-abeilain groups (which means by the way that there exist matrices that are not "square-rootable," since every group is a matrix group).
On the other hand, as far as the definition goes, sqrt(A) just means the element (we have to assume that such an element exists and is unique I guess) such that sqrt(A)*sqrt(A) = A. In other words, I think for non-commutatives, we don't have to worry about the silly assumption nonsense and we can just say sqrt(A**3) == sqrt(A)**3 == A**(3/2) == A*sqrt(A).
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