Well you can assume all sorts of things, but in this case assuming is not nearly as good as concluding something.
If you really mean it when you write "group of vectors," then you are probably assuming that adding two of them together results in a vector that is still in the group, and that the negative version of each vector is in the group, so that the addition operation is the vector operation we all know.
In that case, you would be right that the set $V$ is a subgroup of $R^4$. That's exactly the definition of a subgroup of a group: a subset that is closed under the operation of the containing group, and contains inverses to its own elements.
It is not necessarily a subspace, because there is the possibility it does not contain all scalar multiples of its elements.
Writing "$V$ is a subgroup of $R^4$ and therefore of $R$" does not make much sense, though, since $R^4$ is not a subset of $R$.