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Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$.

Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\mathfrak{S}_E$ generated by $\zeta$. We say that $\zeta$ is a cycle if, under the operation of $\overline{\zeta}$ on $E$, there exists a unique orbit which is not reduced to a single element. This orbit is called the support of $\zeta$ and denoted by $\text{supp}(\zeta)$. The order of $\zeta$ is the cardinality of its support.

Here the operation of $\mathfrak{S}_E$ on $E$ is the identity mapping. Thus for $\sigma \in \mathfrak{S}_E$ and $x \in E$, we define $\sigma.x = \sigma(x)$.

More precisely (correct me if I'm wrong) :

$\zeta$ is a cycle if and only if there exists a unique $x \in E$ such that $$\text{Card}\Big(\overline{\zeta}.x\Big) > 1,$$ and then $\text{supp}(\zeta) = \overline{\zeta}.x$.

It is easy to show that $$\text{Card}\Big(\overline{\zeta}.x\Big) > 1 \iff \zeta(x) \ne x.$$ In fact, the condition is necessary, for if $\zeta(x) = x$, then $\zeta^n(x) = x$ for all $n \in \mathbb{Z}$, so $\overline{\zeta}.x = \{x\}$. Conversely, if $\zeta(x) \ne x$, then $\{x,\zeta(x)\} \subseteq \overline{\zeta}.x$.

The book claims that $y \in \text{supp}(\zeta)$ if and only if $\zeta(y) \ne y$. If everything above is true, then $\text{supp}(\zeta) = \{x\}$, where $x$ is the unique element of $E$ whose orbit is not reduced to a single element.

But this is not true, since a transposition is a cycle of order $2$. Therefore something above is incorrect.

Question.$\ \ $ What am I misunderstanding?

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$\zeta$ is a cycle if and only if there exists a unique $x \in E$ such that $$\text{Card}\Big(\overline{\zeta}.x\Big) > 1,$$

No, $x$ is not unique. What is unique is just the orbit $\overline \zeta\cdot x$, as a subset of $E$. If $y$ is another element of that orbit, we have $\overline \zeta\cdot x = \overline\zeta \cdot y$, yet $x\ne y$.

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That's a rather badly phrased definition. I'd interpret "a unique orbit which is not reduced to a single element" to mean "a unique orbit which doesn't consist of a single element". Then the definition makes sense.

I don't see a basis for your interpretation in the text (and it certainly doesn't correspond to the definition of a cycle). Despite its unnecessary wordiness, the text clearly refers to an orbit as unique and not, as you do, to an element.

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The definition said unique orbit, not unique element $x$ of $E$.

Start with an element of $x\in E$. The orbit is $\{x, \zeta.x, \zeta^2. x, \zeta^3. x, \ldots\}$, and iteration ultimately brings you back to $x$.

You could have started with the element $y=\zeta^3.x$ and then the orbit would be $\{y, \zeta.y, \zeta^2. y, \zeta^3. y, \ldots\}$ and it would list the same elements in a different order.

So the orbit is unique, but the element you start with in that list is not.

The definition doesn't seem like the most clearly written definition I've ever seen.

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