$(34)(123)(456)$ is a cycle. True or False?

Question

I know this is basics, and I understand that $(34)(123)(456)$ is a product of cycles which, I found: $(124563)$.
But somehow, I was lost. How do I know if it is indeed a cycle? OR if it isn't? Any help would be appreciated.

Answer 1

These are getting a bit long for comments, so:

I'm not sure what you mean by 'all elements have been kept track'. For example, if $\Omega = \{1,2,3,4,5\}$, and $\sigma = (123)(45)$ acts on $\Omega$, then all elements of $\Omega$ are accounted for ('kept track'?), but $\sigma$ cannot be expressed as a single cycle.

In general, I suppose to check if a permutation $\sigma$ acting on a set $\Omega$ can be expressed as a single cycle, you need to check if it acts transitively on $\Omega \setminus Fix(\sigma)$. This means that for any two elements $\alpha, \beta \in \Omega$ that are not fixed by $\sigma$, there exists $n\geq 1$ such that $\alpha^{\sigma^n} = \sigma^n (\alpha) = β$.

Given an arbitrary permutation expressed as a product of cycles:

$$\sigma=(\alpha_1 \ldots \alpha_{r_1})(\beta_1 \ldots \beta_{r_2})\ldots(\gamma_1 \ldots \gamma_{r_k})$$

if you want to check whether this can be written as a single cycle, you should first check to see whether the cycles are disjoint or not. If we have $\alpha_i = \beta_j$ say, then the cycles are not disjoint and there is a simpler expression. If however all elements of $\Omega$ appearing in the cycles are distinct, then the cycles are disjoint and this is the expression of $\Omega$ in disjoint cycle notation.

Answer 2

$$ (34)(123)(456) = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 4 & 3 & 5 & 6 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 3 & 1 & 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 & 5 & 6 & 4 \end{pmatrix} $$ Now compose from right to left to get $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 & 1 & 5 & 6 & 3 \end{pmatrix} = (124563) $$ So what you've got appears to be correct - and yes, it is a cycle!