Notation:Let $F_2$ be the free group on generators $x,y$.
Denote the profinite completion of group $G$ by $\hat{G} = \underset{N}{\varprojlim} G/N$.
For any group homomorphism $$ \begin{aligned} &\widehat{F}_{2} \rightarrow G \\ &x, y \mapsto a, b \end{aligned} $$ we write $f(a, b)$ for the image of $f \in \widehat{F}_{2}$. For example:
- under id : $\widehat{F}_{2} \rightarrow \widehat{F}_{2}$, we have $f=f(x, y)$;
- under the map $\widehat{F}_{2} \rightarrow \widehat{F}_{2}$ exchanging the generators $x$ and $y$, we have $$ f=f(x, y) \mapsto f(y, x) $$ Definition 1. The Grothendieck-Teichmüller group $\widehat{G T}$ is the group of pairs $(\lambda, f) \in \widehat{\mathbb{Z}}^{*} \times \widehat{F}_{2}^{\prime}$ such that $x \mapsto x^{\lambda}$ and $y \mapsto f^{-1} y^{\lambda} f$ induces an automorphism of $\widehat{F}_{2}$, and such that
(I) $f(x, y) f(y, x)=1$
(II) $f(x, y) x^{m} f(z, x) z^{m} f(y, z) y^{m}=1$ where $x y z=1$ and $m=(\lambda-1) / 2$,
(III) (5-cycle relation) $f\left(x_{34}, x_{45}\right) f\left(x_{51}, x_{12}\right) f\left(x_{23}, x_{34}\right) f\left(x_{45}, x_{51}\right) f\left(x_{12}, x_{23}\right)=1$ in $\widehat{\Gamma}_{0,5}$, where $x_{i j}$ is the Dehn twist along a loop (on a sphere with 5 numbered marked points) surrounding points $i$ and $j$.
Definition 2. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an automorphism $F_{f}$ of $\hat{F}_{2}$, and which furthermore satisfy the following three relations:
$f\left(a_{2}^{2}, a_{1}^{2}\right) f\left(a_{1}^{2}, a_{2}^{2}\right)=1$ in $\hat{\Gamma}_{1}^{1}$, where $\alpha_{1}$ and $\alpha_{2}$ are as in figure $1(\mathrm{a})$;
$f\left(b_{3}, b_{1}\right) f\left(b_{2}, b_{3}\right) f\left(b_{1}, b_{2}\right)=1$ in $\hat{\Gamma}_{0}^{4}$, where $\beta_{1}, \beta_{2}$ and $\beta_{3}$ are as in figure $1(\mathrm{~b})$;
$f\left(b_{3}, b_{4}\right) f\left(b_{5}, b_{1}\right) f\left(b_{2}, b_{3}\right) f\left(b_{4}, b_{5}\right) f\left(b_{1}, b_{2}\right)=1$ in $\hat{\Gamma}_{0}^{5}$, where the $\beta_{i}$ are as in figure $1(\mathrm{c})$.
In this definition, $f(a, b)$ denotes the image of $f$ under a homomorphism of $\hat{F}_{2}$ into some profinite group $G$ sending $x \mapsto a$ and $y \mapsto b$. The set $\widehat{G T}^{1}$ is made into a group by defining the multiplication law $f \cdot g$ to be given by composition of the automorphisms. In other words, if $F_{f}$ and $F_{g}$ denote the automorphisms of $\hat{F}_{2}$ associated to $f$ and $g \in \widehat{G T}^{1}$ then the automorphism $F_{g f}$ is defined to be $F_{g} \circ F_{f}$, so that we have $g \cdot f=g F_{g}(f)$
if we set $\lambda=1$ in Definition 1 then we have Definition of $\widehat{G T}^{1}$. how we can proof Definition 1 and Definition 2 is equivalent definition ( for $\lambda=1$) ?