My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group? I know that there are abelian groups that we can not endow it with a structure of vector space over any field, for example $\mathbb{Z}_{6}$. But if an abelian group has a structure of a vector space over a field $\mathbb{F}$, is there an upper bound for ways we can define a different structure? Such as the number of automorphisms of the field $\mathbb{F}$. I am conjecturing according to the answer given before.
number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group
4
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linear-algebra
field-theory
abelian-groups
1 Answers
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A $k$-vector space structure on an abelian group $A$ is nothing more than a ring homomorphism $k \to \text{End}(A)$. From this it follows that if k' is another field and k' \to k a field homomorphism, then the composition k' \to k \to \text{End}(A) gives $A$ the structure of a k'-vector space. In particular, k' may be any subfield of $k$, and we can also compose with any automorphism of k'.
More importantly, since a ring homomorphism $k \to \text{End}(A)$ is necessarily injective, the set of vector space structures on $A$ can be identified (non-canonically) with the set of pairs $(\text{subring of } \text{End}(A) \text{ that is a field } k, \text{automorphism of } k).$
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0@spohreis: okay, so replace "a field $k$" above with "isomorphic to $F$." It's still not as simple as counting the automorphisms of $F$ because there may actually be different subrings of $\text{End}(A)$ isomorphic to $F$ (for example there exist fields which embed properly into themselves). – 2012-01-26