Elementary abelian group

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In group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group, where every nontrivial element has order p, where p is a prime; it is a particular kind of p-group.[1][2] The case where p = 2, i.e. 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 infinte case.)

Presently, in the rest of this article, these groups are assumed finite.

Examples and properties[edit]

  • 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 mod 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 xy = (xy)2 = (xy)-1 = y-1x-1 = y2x2 = 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.
 (\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[edit]

Suppose V \cong (Z/pZ)n is an elementary abelian group. Since Z/pZ \cong Fp, the finite field of p elements, we have V = (Z/pZ)n \cong 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 \overset{\sim}{\to} (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.

Automorphism group[edit]

As a 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.

A generalisation to higher orders[edit]

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 (pe,pe,...,pe) i.e. the direct product of n isomorphic groups of order pe.

Related groups[edit]

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[edit]

References[edit]

  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.