Subgroups which are not subspaces 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?

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.
