12 Jan
2019
12 Jan
'19
9:59 p.m.
On Sat, Jan 12, 2019 at 9:34 PM Dan Asimov <dasimov@earthlink.net> wrote:
Just something amusing I'm learning about — For each integer let F_n be the free group on n generators, nonabelian for n >= 2.°
These groups are utterly natural yet quite intriguing. Like the fact that F_2 contains an isomorphic copy of F_n for any n, even if n = aleph_0.
My favorite fact about free groups is that these are the *only* subgroups of F_n; that is, any subgroup of a free group is free. If you think of this as a fact about algebra, I don't see how you'd go about proving it. But if you think of it as a fact about topology, it has a very simple explanation: The fundamental group of a graph is a free group. A covering space of a graph is a graph. QED. Andy