When can $A$ an abelian group be made into a vector space over $\Bbb{F}_p$? Let $\Bbb{F}_p$ be the finite field of integers modulo $p, p$ a prime, let $A$ be an abelian group. Precisely when can $A$ be made into a vector space over $\Bbb{F}_p$?
 A: I'll try to give a full answer based on the comments
An abelian group $A$ is a $\mathbb{Z}-$module and $ann_{\mathbb{Z}}A=(d)$ for some $d\in\mathbb{Z}$.  
I'll show that $A$ is a $\mathbb{F}_p$ vector space iff $(p)\subset (d)\Leftrightarrow d|p\Leftrightarrow p=d$


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*Let $ann_{\mathbb{Z}}=(p)$. Then we can define a $\dfrac{\mathbb{Z}}{p\mathbb{Z}}-$module structure as follows $$\dfrac{\mathbb{Z}}{p\mathbb{Z}}\times A\to A$$
with $(z+p\mathbb{Z},x)\mapsto zx$. This is well-defined : if $(z+p\mathbb{Z},x)=(z'+p\mathbb{Z},x')\Rightarrow x=x',z-z'\in p\mathbb{Z}\Rightarrow (z-z')x=0\Rightarrow zx=z'x'.$ This way $A$ becomes a $\mathbb{F}_p-$module (=$\mathbb{F}_p$-vector space)

*Let $A$ an $\mathbb{F}_p-$vector space (=$\dfrac{\mathbb{Z}}{p\mathbb{Z}}$-module). Then consider the ring homomorphism $$f:\mathbb{Z}\to \dfrac{\mathbb{Z}}{p\mathbb{Z}}$$ with $f(z)=[z]_p$. Then we can see $A$ as a $\mathbb{Z}-$module with multiplication $$z\cdot a=f(z)a,\quad \forall a\in A,z\in \mathbb{Z}$$. Then $$z\cdot a=0\Leftrightarrow f(z)a=0\Leftrightarrow f(z)=[0]_p\Leftrightarrow p|z$$ hence $ann_{\mathbb{Z}}=(p)$

