$d$ to $1$ map, cyclic group and cosets We know that $\mathbb{F}_p^{\times}$ is a cyclic group. Let $g$ be a prmitive root mod $p$, then the kernel of the map $\varphi:\mathbb{F}_p^{\times}\rightarrow \mathbb{F}_p^{\times}$ defined by $\varphi (x)=x^k$ has order $d= gcd (p-1,k)$. Therefore the map is $d$ to $1$ and we have $d$ ditinct cosets: $H, gH ,g^2H, \cdots,g^{d-1}H$, where $H=\{x^k \pmod{p}| x\in\mathbb{Z}/p\mathbb{Z}\}$. i.e. any element in $\mathbb{Z}/p\mathbb{Z}$  is either of the form $x^k,gx^k,\cdots,g^{d-1}x^k.$
I don't understand what it says after "therefore the map is $d$ to $1$..." Can anyone kindly explain? thank you.

Hints (using your notation):
$$\ker\phi:=\left\{x\in\Bbb F_p^*\;;\;x^k=1\right\}=:H\iff [G:H]=\left|G/H\right|=\frac{p-1}{|H|}=d$$
Thus, we have $\,d\,$ elements in the quotient group $\,G/H\,$, and we know this quotient group is cyclic, so
$$\exists\,g\in G\;\;s.t.\;\;G/H=\langle\,gH\,\rangle=\left\{H\,,\,gH\,,\,g^2H\,\ldots,g^{d-1}H\right\}\implies$$
$$\forall\,y\in G\;\exists\,!k_y\in\Bbb N\,,\,0\le k_y\le d-1\,,\wedge\,\exists!h_y\in H\;\;s.t.\;\;\;y=g^{k_y}h_y$$
