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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, an elementary abelian group (or elementary abelian ''p''-group) is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
in which every nontrivial element has order ''p''. The number ''p'' must be
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

prime
, and the elementary abelian groups are a particular kind of ''p''-group. The case where ''p'' = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group. Every elementary abelian ''p''-group is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over the
prime field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with ''p'' elements, and conversely every such vector space is an elementary abelian group. By the
classification of finitely generated abelian groups In abstract algebra, an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...
, or by the fact that every vector space has a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
, every finite elementary abelian group must be of the form (Z/''p''Z)''n'' for ''n'' a non-negative integer (sometimes called the group's ''rank''). Here, Z/''p''Z denotes the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
of order ''p'' (or equivalently the integers mod ''p''), and the superscript notation means the ''n''-fold
direct product of groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. In general, a (possibly infinite) elementary abelian ''p''-group is a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of cyclic groups of order ''p''. (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.) Presently, in the rest of this article, these groups are assumed
finite Finite is the opposite of infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infin ...
.


Examples and properties

* The elementary abelian group (Z/2Z)2 has four elements: . Addition is performed componentwise, taking the result modulo 2. For instance, . This is in fact the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
. * In the group generated by the
symmetric difference In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
on a (not necessarily finite) set, every element has order 2. Any such group is necessarily abelian because, since every element is its own inverse, ''xy'' = (''xy'')−1 = ''y''−1''x''−1 = ''yx''. Such a group (also called a Boolean group), generalizes the Klein four-group example to an arbitrary number of components. * (Z/''p''Z)''n'' is generated by ''n'' elements, and ''n'' is the least possible number of generators. In particular, the set , 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 In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...
. :: (\mathbb Z/p\mathbb Z)^n \cong \langle e_1,\ldots,e_n\mid e_i^p = 1,\ e_i e_j = e_j e_i \rangle


Vector space structure

Suppose ''V'' \cong (Z/''p''Z)''n'' is an elementary abelian group. Since Z/''p''Z \cong F''p'', the
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of ''p'' elements, we have ''V'' = (Z/''p''Z)''n'' \cong F''p''''n'', hence ''V'' can be considered as an ''n''-dimensional
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over the field F''p''. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism ''V'' \overset (Z/''p''Z)''n'' corresponds to a choice of basis. 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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
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''·''g'' = ''g'' + ''g'' + ... + ''g'' (''c'' times) where ''c'' in F''p'' (considered as an integer with 0 ≤ ''c'' < ''p'') gives ''V'' a natural F''p''-module structure.


Automorphism group

As a vector space ''V'' has a basis as described in the examples, if we take to be any ''n'' elements of ''V'', then by
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
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 In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
) 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
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

automorphism
s of ''V'' we have Aut(''V'') = = GL''n''(F''p''), the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of ''n'' × ''n'' invertible matrices on F''p''. The automorphism group GL(''V'') = GL''n''(F''p'') acts transitively on ''V \ '' (as is true for any vector space). This in fact characterizes elementary abelian groups among all finite groups: if ''G'' is a finite group with identity ''e'' such that Aut(''G'') acts transitively on ''G \ '', then ''G'' is elementary abelian. (Proof: if Aut(''G'') acts transitively on ''G \ '', then all nonidentity elements of ''G'' have the same (necessarily prime) order. Then ''G'' is a ''p''-group. It follows that ''G'' has a nontrivial
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
, which is necessarily invariant under all automorphisms, and thus equals all of ''G''.)


A generalisation to higher orders

It can also be of interest to go beyond prime order components to prime-power order. Consider an elementary abelian group ''G'' to be of ''type'' (''p'',''p'',...,''p'') for some prime ''p''. A ''homocyclic group'' (of rank ''n'') is an abelian group of type (''m'',''m'',...,''m'') i.e. the direct product of ''n'' isomorphic cyclic groups of order ''m'', of which groups of type (''pk'',''pk'',...,''pk'') are a special case.


Related groups

The extra special groups are extensions of elementary abelian groups by a cyclic group of order ''p,'' and are analogous to the
Heisenberg group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
.


See also

* Elementary group *
Hamming space In statistics and coding theory, a Richard Hamming, Hamming space is usually the set of all 2^N binary strings of length ''N''. It is used in the theory of coding signals and transmission. More generally, a Hamming space can be defined over any al ...


References

{{Reflist Abelian group theory Finite groups P-groups