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.

$G$ can be seen as a vector space over its subfield $\{0,1,2\}$, of dimension 2; this is the usual argument that a field has a prime power as its cardinality.
This shows 1) holds. This rules out 2), as $\mathbb{Z} / 9\mathbb{Z}$ has an additive element of order 9, while the first one has not.
For any field $F$, $F \setminus \{0\}$ is cyclic (as a multiplicative group) (theorem), which rules out 3. Check that 4 is indeed true.
