1. ## Left Representation

Let G be a finite group and let g $\displaystyle \in$ G. Let m be the order of g. Prove that in the representation $\displaystyle G$$\displaystyle \to$$\displaystyle S_{|G|}$ (given by left multiplication) every cycle in the cycle decomposition of the image of g has length m.

2. ## Re: Left Representation

Originally Posted by kierkegaard
Let G be a finite group and let g $\displaystyle \in$ G. Let m be the order of g. Prove that in the representation $\displaystyle G$$\displaystyle \to$$\displaystyle S_{|G|}$ (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 $\displaystyle x\in G$ then one has the $\displaystyle m$-cycle $\displaystyle (x,gx,\cdots,g^{m-1}x)$, then selecting $\displaystyle y\in G-\{x,gx,\cdots,g^{m-1}x)$ one has the $\displaystyle m$-cycle $\displaystyle (y,gy,\cdots,g^{m-1}y)$. I think you get the idea now.

3. ## Re: Left Representation

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

how is this relevant?