## Thursday, May 10, 2012

### An elementary exercise in group actions

Let $$G$$ be a group, $$T$$ a normal subgroup, and suppose that $$G$$ acts transitively on a set $$X$$. Then the induced action of $$T$$ on $$X$$ may not be transitive; let $$T \setminus X$$ denote the orbit space. Show that the quotient group $$G/T$$ acts well-definedly and transitively on $$T \setminus X$$.

I plan to do this somewhat regularly... I know that I'm terrible about keeping up a store of problems and examples, which is a key component of mathematical pedagogy.