I'm reading a paper on Transfer and Fusion in finite groups, and I've come across a lemma in which we:
" Let $x \in G$ and let $n_1 , ... , n_r$ be the sizes of the cycles of $x$ on $\Omega$ ... "
Previously it was defined that $\Omega = G/H$ where $H < G$ .
I am confused about the terminology being used here. What does it mean for x to have a cycle on $\Omega$?
My best guess so far is that it has to do with how x interacts with each element of $\Omega$, taken as a coset of $H$. But, assuming my intuition is correct, exactly what this interaction is eludes me.
$x$ is taken to act on $\Omega$ by permuting the cosets $\alpha \in \Omega$. The cycles of $x$ on $\Omega$ are simply the cycles of elements $\alpha$ of $\Omega$ under the action of (in this case right) multiplication of $x$.