# For a finite group $G$, disjoint cycles form a right regular representation $T_g (x) = xg$ of $G$, each cycle having length $|g|.$

I want clarification on a question from Gallian's book contemporary abstract algebra, 8th edition, number 62 of section 6. The statement to be proven is:

Let $G$ be a finite group. Then in disjoint cycles form of the right regular representation $T_g (x) = xg$ of $G$, each cycle has length $|g|$.

I'm not quite sure what I'm supposed to show here. A restatement might be all I need to start trying things. I don't really know what is meant by right regular representation. They don't really say anything about what $x$ and $g$ are here. I'm not looking for how to prove the statement, just what the statement means.

## 2 Answers

HINT: How many different elements can you get by applying $T_g$ recursively?

I think the author means the following: Let $g \in G$ then $\left\langle g \right\rangle$ is a cyclic group and the orbit of an element under the regular representation is a cycle of order $|g|$, and does not depend on how $g$ decomposes as cycles as in the example $g = (1,2)(3,4,5,6)$.