Thread: Left Representation

Let G be a finite group and let g G. Let m be the order of g. Prove that in the representation (given by left multiplication) every cycle in the cycle decomposition of the image of g has length m.

Originally Posted by kierkegaard

Let G be a finite group and let g G. Let m be the order of g. Prove that in the representation (given by left multiplication) every cycle in the cycle decomposition of the image of g has length m.

The basic idea is that if one selects then one has the -cycle , then selecting one has the -cycle . I think you get the idea now.

let H = <g>, and be the image of g in the representation. what is the action of on the right cosets Hx?

how is this relevant?