following equalities hold: $$x^{-1}\cdot(x\cdot y)=y ~~~\hbox{and}~~~(x\cdot y)\cdot
y^{-1}=x.$$

In a groupoid $(G, G_{0})$ for any $ u\in G_{0}$, the set $~G(u):=\alpha^{-1}(u)\cap\beta^{-1}(u)=\{ x\in G~|~
\alpha(x)=\beta(x)=u~\}$ is a group under the restriction of the partial multiplication $~ m~$ to $ G(u),$ called the [*isotropy group at*]{} $u$ of $G$.

([@ivan02])

*Let $~( G, \alpha, \beta, m, \iota , G_{0})~$ be a groupoid. Then:*

$(i)~~~~~\alpha \circ \iota = \beta,~~ \beta \circ \iota =
\alpha~~ \hbox{and}~~ \iota \circ \iota = Id_{G}.$

$(ii)~~~\varphi : G(\alpha(x)) \to G(\beta(x)),~ \varphi (z):=
x^{-1} z x $ is an isomorphism of groups.

$(iii)~ $ If $ (G, G_{0} ) $ is transitive, then all isotropy groups are isomorphes.

A [*group bundle*]{} is a groupoid $~( G, G_{0})~$ with the property that $~\alpha (x) = \beta (x)~$ for all $~x\in G .$ Moreover,a group bundle is the union of its isotropy groups $~G(u)
= \alpha^{-1}(u), u \in G_{0}~$ (here, two elements may be composed iff they lie in the same fiber $\alpha^{-1}(u)$ ).

If $~( G,\alpha,\beta, G_{0})~$ is a groupoid then $~Is(G): = \{
x\in G~|~\alpha (x)=\beta (x) \}~$ is a group bundle, called the [*isotropy group bundle*]{} of $~G.~$

[*(i)*]{}  Any group $ G $ having $ e $ 