Let $(F,+,\cdot)$ is the finite field with $9$ elements. Then which of the following are true? Let $(F,+,\cdot)$ is the finite field with $9$ elements. Let $G=(F,+)$ and $H=(F-\{0\}, \cdot)$ denote the underlying additive and multiplicative groups respectively. Then, 


*

*$G\cong (\mathbb{Z/3Z})\times(\mathbb{Z/3Z})$

*$G\cong\mathbb{Z/9Z}$

*$H\cong(\mathbb{Z/2Z})\times(\mathbb{Z/2Z})\times(\mathbb{Z/2Z})$

*$G\cong (\mathbb{Z/3Z})\times(\mathbb{Z/3Z})$ and $H\cong\mathbb{Z/8Z}$


I guess $1$ and $2$ and $4$ are true but not convincingly. Can anybody help me out? Hints will be fine.

Note that the characteristic of the field is 3.So the field F is the splitting field of  $x^{3^2}-x$ on $F_3$. I doubt whether (1) holds. Since (0,1) does not have a reverse.
Update: Since G is an ADDITIVE group I think (1) holds.
