{\cal SG}(A,X) =\{ f : A \to A~|~ \emptyset\neq A\subseteq X,~ f~ \hbox{is bijective}~ \}
 ~$ and\
$~ G_{0}:= \{ Id_{A}~|~\emptyset \neq A\subseteq X \} $, where $ Id_{A} $ is the identity map on $ A.$\
Let $ G_{(2)}:= \{(f,g)\in G\times G | D(f)=D(g) \} $, where $ D(f)$ denotes the domain of $ f $. The structure functions $~ \alpha, \beta : G \to G_{0},~ \iota : G
\to G~ $ and the multiplication $~ m : G_{(2)} \to G ~$ are given by:\
$ \alpha(f):= Id_{D(f)},~~ \beta(f):= Id_{D(f)},~~ \iota(f):=
f^{-1}~~ $ and $~~ m(f,g):= f\circ g $.

Then $ (G, G_{0}) $ is a groupoid, called the [*groupoid of bijective functions from the subsets $ A $ of $ X $ onto $ A $*]{} or the [*symmetry groupoid of the set $X$*]{}.

The isotropy group at $ u= Id_{A}$ is the symmetry group of the set $A$, i.e. $ G(u) = \{ f : A \to A~|~ f~ \hbox{is bijective}~ \}$.

In particular, the symmetry groupoid of a finite set $ X = \{
x_{1}, x_{2},\ldots, x_{n} \}, $ is called the [*symmetry groupoid of degree $ n $*]{} and is denoted by $ {\cal SG}_{n}$. 