Cardinality of a group of permutation Let $S$ be an infinite set of cardinality $\alpha$ and $G$ be a subgroup of $Sym(S)$. Let $\sigma(g)=\{s\in S \mid sg\neq s\}$ for each $g\in G$ and define $$Sym(S,\, \alpha)=\{g\in Sym(S)\mid |\sigma(g)|<\alpha\}.$$ I was trying to establish the cardinality of $Sym(S,\, \alpha)$.
What is the cardinality of $Sym(S,\, \alpha)$, and how to prove it?

Old Question
I argued pretty much as follows:

*

*Every element $g\in Sym(S,\, \alpha)$ may be constructed in the following way: take a subset $T$ of cardinality strictly smaller than
$\alpha$ and consider a permutation of $S$ which fixes all the
elements of $S\backslash T$.


*How many subsets of cardinality strictly less than $\alpha$ are there? For each cardinality $\beta <\alpha$ I find
$\alpha^\beta=\alpha$ (this is not true) subsets of cardinality
$\beta$. Since there are $\alpha$ cardinalities before $\alpha$ the
total number is $\alpha$.


*For each $L$ of this subsets I need to consider a permutation. Since $|L|<\alpha$ the set $|Sym(L)|=2^{|L|}\leq \alpha$.
So, at the end, it follows that $|Sym(S,\, \alpha)|=\alpha$.
Are my arguments correct?

 A: The cardinality of $Sym(S,\alpha)$ is $\alpha^{<\alpha}$: that is, it is the supremum of the cardinals $\alpha^\beta$ where $\beta$ ranges over all cardinals less than $\alpha$.
First, I claim that $|Sym(S,\alpha)|\geq\alpha^{<\alpha}$.  Indeed, fix a cardinal $\beta<\alpha$ and identify $S$ with the set $\beta\times(\alpha+1)$.  For any function $f:\beta\to\alpha$, consider the permutation of $S=\beta\times(\alpha+1)$ which swaps $(x,\alpha)$ and $(x,f(x))$ for each $x\in \beta$ and is otherwise the identity.  The support of this permutation has cardinality $\beta$, and so this defines an injection from the set of functions $\beta\to\alpha$ to $Sym(S,\alpha)$.  Thus $|Sym(S,\alpha)|\geq\alpha^\beta$.  Since $\beta<\alpha$ was arbitrary, this means $|Sym(S,\alpha)|\geq\alpha^{<\alpha}$.
Conversely, I claim $|Sym(S,\alpha)|\leq\alpha^{<\alpha}$.  Indeed, if $g\in Sym(S,\alpha)$, then $g$ is uniquely determined by a pair of maps $\beta\to S$ for some $\beta<\alpha$: one map which is a bijection from $\beta$ to the support of $g$, and one map which then says where each element of the support is sent by $g$.  There are $\alpha^\beta\cdot\alpha^\beta=\alpha^\beta$ pairs of functions $\beta\to S$.  This shows that $$|Sym(S,\alpha)|\leq\sum_{\beta<\alpha}\alpha^\beta.$$  Now $\alpha^\beta\leq \alpha^{<\alpha}$ for each $\beta<\alpha$, and there are at most $\alpha$ cardinals less than $\alpha$.  So the sum above is a sum of at most $\alpha$ terms, each of which is at most $\alpha^{<\alpha}$.  Thus $$|Sym(S,\alpha)|\leq\alpha^{<\alpha}\cdot\alpha=\alpha^{<\alpha}.$$
