Isomorphism of Groups (C.S.I.R)

Question

Suppose $(F , +, .)$ is the finite Field with $9$ elements. Let $G = (F , +)$ and $H = (F \setminus \{0\}, .)$ denotes the underlying additive and multiplicative groups respectively. Then which are TRUE ?

1. $ G \cong \mathbb Z_3 \times \mathbb Z_3$.

2. $G \cong \mathbb Z_9$.

3. $ H \cong \mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$.

4. $ G \cong \mathbb Z_3 \times \mathbb Z_3$ and $ H \cong \mathbb Z_8$

I know that $H$ is a cyclic group and $o(G) = 3^2$, So $G \cong \mathbb Z_9$ or $ G \cong \mathbb Z_3 \times \mathbb Z_3$.

Please tell me about G.

Thank you

Answer 1

Hint: If $p$ is the characteristic of the field (I assume $F$ is a field here), then $px = 0$ for all $x\in G$.

Answer 2

A field of order $p^n$ is a vector space of dimension $n$ over $\mathbb{Z}_p$. As a vector space $F_{p^n} \cong \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$, $n$ times. Hence here $$G \cong \mathbb{Z}_3 \times \mathbb{Z}_3.$$