Two fun problems November 26, 2008Posted by Noah Snyder in fun problems, group theory, linear algebra, subfactors.
One of the points of this blog is for us to share the little problems we’d be discussing at tea if we were all still in Berkeley. Here are two that came up in the last couple weeks.
As we all know, you can never know too much linear algebra. So here’s a fun little linear algebra exercise that Dave Penneys asked us over beers on friday: “Which matrices have square roots?”
The second question I don’t know the answer to, but I haven’t looked too hard. The other week Penneys and I were trying to compute an example in subfactors and stumbled on the following interesting question about infinite groups (somewhat reminiscent of this old post). When can you find a group G and a proper inclusion G->G such that the image is finite index?
There’s the obvious example Z. But once you start adding adjectives it starts getting tricky. We were looking for a finitely generated group all of whose nontrivial conjugacy classes are infinite. If only I knew more geometric group theory…