Fpn, hence V can be considered as an n-dimensional vector space over the field Fp.
Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism V
That is, c⋅g = g + g + ... + g (c times) where c in Fp (considered as an integer with 0 ≤ c < p) gives V a natural Fp-module structure.
As a finite-dimensional vector space V has a basis {e1, ..., en} as described in the examples, if we take {v1, ..., vn} to be any n elements of V, then by linear algebra we have that the mapping T(ei) = vi extends uniquely to a linear transformation of V. Each such T can be considered as a group homomorphism from V to V (an endomorphism) and likewise any endomorphism of V can be considered as a linear transformation of V as a vector space.
The automorphism group GL(V) = GLn(Fp) acts transitively on V \ {0} (as is true for any vector space).
(Proof: if Aut(G) acts transitively on G \ {e}, then all nonidentity elements of G have the same (necessarily prime) order.