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In mathematics, specifically in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, an elementary abelian group (or elementary abelian ''p''-group) is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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 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 because the only ways ...
, 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 and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
over the
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
with ''p'' elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
, 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 of order ''p'' (or equivalently the integers mod ''p''), and the superscript notation means the ''n''-fold
direct product of groups In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is o ...
. In general, a (possibly infinite) elementary abelian ''p''-group is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
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. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
.


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 with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one ...
. * In the group generated by the
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. T ...
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. :: (\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, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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 and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
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 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 matrices. ...
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 gr ...
) 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, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of ''V'' we have Aut(''V'') = = GL''n''(F''p''), the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ...
, 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 group In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime ''p'' and positive integer ''n'' there are exactly two (up to isomorphism) extraspeci ...
s are extensions of elementary abelian groups by a cyclic group of order ''p,'' and are analogous to the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ' ...
.


See also

*
Elementary group In algebra, more specifically group theory, a ''p''-elementary group is a direct product of a finite cyclic group of order relatively prime to ''p'' and a ''p''-group. A finite group is an elementary group if it is ''p''-elementary for some prime ...
*
Hamming space In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) 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 Ham ...


References

{{Reflist Abelian group theory Finite groups P-groups