TITLE: What will the underlying group of a field be isomorphic to?
QUESTION [5 upvotes]: Let $(F,+,.)$ be a finite field with 9 elements. Let $G=(F,+)$ and $H=(F\setminus \{0\},.)$ denote the underlying additive and multiplicative groups. Then what will $G$ and $H$ be isomorphic to?
We know that any finte abelian group is a direct product of cyclic group thus either $G$ is isomorphic to $\mathbb Z_9$ or $\mathbb Z_3\times \mathbb Z_3$ and $H$ is isomorphic to either $\mathbb Z_8$ or $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Since a field can have no zero divisor theus $G$ willbe isomorphic to $\mathbb Z_3\times \mathbb Z_3$
But I can't conclude what $H$ will be isomorphic to. Any help

REPLY [3 votes]: Actually any  finite subgroup of the multiplicative group of a field (whether the field itself is finite or not) is cyclic. In the present case, 
$$\mathbf F_9^{\times}\simeq \mathbf Z/8\mathbf Z$$