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.

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