Elementary abelian group

Elementary abelian group

In group theory an elementary abelian group is a finite abelian group, where every nontrivial element has order "p" where "p" is a prime.

By the classification of finitely generated abelian groups, every elementary abelian group must be of the form

:("Z"/"pZ")"n"

for "n" a non-negative integer. Here "Z/pZ" denotes the cyclic group of order "p" (or equivalently the integers mod "p"), and the notation means the "n"-fold Cartesian product.

Examples and properties

* The elementary abelian group ("Z"/2"Z")2 has four elements: { [0,0] , [0,1] , [1,0] , [1,1] }. Addition is performed componentwise, taking the result mod 2. For instance, [1,0] + [1,1] = [0,1] .

* ("Z"/"pZ")"n" is generated by "n" elements, and "n" is the least possible number of generators. In particular the set {"e"1, ..., "e""n"} where "e""i" has a 1 in the "i"th component and 0 elsewhere is a minimal generating set.

* Every elementary abelian group has a fairly simple finite presentation.

:: ("Z"/"pZ")"n" cong < "e"1, ..., "e""n" | "e""i""p" = 1, "e""i""e""j" = "e""j""e""i" >

Vector space structure

Suppose "V" = ("Z"/"pZ")"n" is an elementary abelian group. Since "Z"/"pZ" cong "F""p", the finite field of "p" elements, we have "V" = ("Z"/"pZ")"n" cong "F""p""n", hence "V" can be considered as an "n"-dimensional vector space over the field "F""p".

To the observant reader it may appear that F"p""n" 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 "F""p" scalar multiplication. That is, "c"&middot;"g" = "g" + "g" + ... + "g" ("c" times) where "c" in "F""p" (considered as an integer with 0 &le; "c" < "p") gives "V" a natural "F""p"-module structure.

Automorphism group

As a vector space "V" has a basis {"e"1, ..., "e""n"} as described in the examples. If we take {"v"1, ..., "v""n"} to be any "n" elements of "V", then by linear algebra we have that the mapping "T"("e""i") = "v""i" 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.

If we restrict our attention to automorphisms of "V" we have Aut("V") = { "T" : "V" -> "V" | ker "T" = 0 } = GL"n"("F""p"), the general linear group of "n" &times; "n" invertible matrices on F"p".

References


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