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Elementary abelian group

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In mathematics, specifically in group theory, an elementary abelian group is an abelian group in which all elements other than the identity have the same order. This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group.[1][2] A group for which p = 2 (that is, an elementary abelian 2-group) is sometimes called a Boolean group.[3]

Every elementary abelian p-group is a vector space over the prime field 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, every finite elementary abelian group must be of the form (Z/pZ)n for n a non-negative integer (sometimes called the group's rank). Here, Z/pZ 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.[2]

In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p.[4] (Note that in the finite case the direct product and direct sum coincide, but this is not so in the infinite case.)

Examples and properties

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  • The elementary abelian group (Z/2Z)2 has four elements: {(0,0), (0,1), (1,0), (1,1)} . Addition is performed componentwise, taking the result modulo 2. For instance, (1,0) + (1,1) = (0,1). This is in fact the Klein four-group.
  • In the group generated by the symmetric difference 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−1x−1 = yx. Such a group (also called a Boolean group), generalizes the Klein four-group example to an arbitrary number of components.
  • (Z/pZ)n is generated by n elements, and n is the least possible number of generators. In particular, the set {e1, ..., en} , where ei has a 1 in the ith component and 0 elsewhere, is a minimal generating set.
  • Every finite elementary abelian group has a fairly simple finite presentation.

Vector space structure

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Suppose V (Z/pZ)n is a finite 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, cg = g + g + ... + g (c times) where c in Fp (considered as an integer with 0 ≤ c < p) gives V a natural Fp-module structure.

Automorphism group

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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.

If we restrict our attention to automorphisms of V we have Aut(V) = { T : VV | ker T = 0 } = GLn(Fp), the general linear group of n × n invertible matrices on Fp.

The automorphism group GL(V) = GLn(Fp) acts transitively on V \ {0} (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 \ {e}, then G is elementary abelian. (Proof: if Aut(G) acts transitively on G \ {e}, 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, which is necessarily invariant under all automorphisms, and thus equals all of G.)

A generalisation to higher orders

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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[5] (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.

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The extra special groups are extensions of elementary abelian groups by a cyclic group of order p, and are analogous to the Heisenberg group.

See also

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References

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  1. ^ Hans J. Zassenhaus (1999) [1958]. The Theory of Groups. Courier Corporation. p. 142. ISBN 978-0-486-16568-4.
  2. ^ a b H.E. Rose (2009). A Course on Finite Groups. Springer Science & Business Media. p. 88. ISBN 978-1-84882-889-6.
  3. ^ Steven Givant; Paul Halmos (2009). Introduction to Boolean Algebras. Springer Science & Business Media. p. 6. ISBN 978-0-387-40293-2.
  4. ^ L. Fuchs (1970). Infinite Abelian Groups. Volume I. Academic Press. p. 43. ISBN 978-0-08-087348-0.
  5. ^ Gorenstein, Daniel (1968). "1.2". Finite Groups. New York: Harper & Row. p. 8. ISBN 0-8218-4342-7.