Subgroups which are not subspaces

Question

Let $p$ be a prime number. Is every subgroup of the abelian group $\Bbb Z_p^2$ a subspace of it as a vector space over $\Bbb Z_p$?

Can it be generalized to all finite fields?

Answer 1

In the case of a prime field: yes. Every proper, non-trivial subgroup is of order $p$ and therefore cyclic. So, if $v$ generates this set, there is an isomorphism $\phi$ determined by $\phi(v) = (1,0)$.

Counterexample in the other case:

For $\mathbb{F}_4$, the subfield $\{0,1\}$ is not a vector subspace of $\mathbb{F}_4$ over $\mathbb{F}_4$.

Since every finite field has prime characteristic, the subgroup generated by $1$ will always be a proper subfield.

Answer 2

This is true of prime fields.

If the order of the field is not a prime, then you can find counterexamples already in One-dimensional spaces, and you should try.