# Automorphism group of disjoint cycle graphs of different lengths

This question is supplementary to another question.

From that question, we know that the automorphism group of the $N$ disjoint cycle graphs of same length $n$ is $S_N \wr D_n$.

My question: What is the Automorphism group of disjoint cycle graphs of different lengths?

My effort: I understand that there will be two cases:

1. More than one cycle graphs with same length
2. Only one cycle graph for each length

Can anyone help me to find for a general solution or special solution to each of these cases?

• There are no automorphisms between cycle graphs of different lengths since there is no bijection between the underlying vertex sets. If your disjoint union looks like $N_1C_{n_1}\sqcup\cdots N_kC_{n_k}$ with $n_i\neq n_j$ for $i\neq j$, you get a direct sum of $S_{N_i}\wr D_{n_i}$ – David Hill Oct 21 '15 at 19:41
• So, the automorphism group is $\sum_i S_{N_i} \wr D_{n_i}$, right? – Omar Shehab Oct 21 '15 at 19:52
• Yes. That is right. – David Hill Oct 21 '15 at 19:52