Vector Space Structure
Suppose V (Z/pZ)n is an elementary abelian group. Since Z/pZ Fp, the finite field of p elements, we have V = (Z/pZ)n 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 (Z/pZ)n corresponds to a choice of basis.
To the observant reader, it may appear that Fpn has more structure than the group V, in particular that it has scalar multiplication in addition to (vector/group) addition. However, V as an abelian group has a unique Z-module structure where the action of Z corresponds to repeated addition, and this Z-module structure is consistent with the Fp scalar multiplication. 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.
Read more about this topic: Elementary Abelian Group
Famous quotes containing the words space and/or structure:
“Stars scribble on our eyes the frosty sagas,
The gleaming cantos of unvanquished space . . .”
—Hart Crane (1899–1932)
“With sixty staring me in the face, I have developed inflammation of the sentence structure and definite hardening of the paragraphs.”
—James Thurber (1894–1961)