Elementary abelian group

<h2 id="examples-and-properties">Examples and properties</h2>
<ul><li>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 <a href="/facts/Klein_four-group/X2oQCYBy">Klein four-group</a>.</li>
<li>In the group generated by the <a href="/facts/Symmetric_difference/W2QST1Qr">symmetric difference</a> 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.</li>
<li>(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.</li>
<li>Every finite elementary abelian group has a fairly simple <a href="/facts/Presentation_of_a_group/f9emPUoI">finite presentation</a>:

(
 
 Z
 
 
 /
 
 p
 
 Z
 
 
 )
 
 n
 
 
 ≅
 ⟨
 
 e
 
 1
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 ∣
 
 e
 
 i
 
 
 p
 
 
 =
 1
 ,
  
 
 e
 
 i
 
 
 
 e
 
 j
 
 
 =
 
 e
 
 j
 
 
 
 e
 
 i
 
 
 ⟩
 
 
 {\displaystyle (\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 }
 
</li></ul>
<h2 id="vector-space-structure">Vector space structure</h2>
Suppose V 
 
 
 
 ≅
 
 
 {\displaystyle \cong }
 
 (Z/pZ)n is a finite elementary abelian group. Since Z/pZ 
 
 
 
 ≅
 
 
 {\displaystyle \cong }
 
 Fp, the <a href="/facts/Finite_field/UkMGJo3m">finite field</a> of p elements, we have V = (Z/pZ)n 
 
 
 
 ≅
 
 
 {\displaystyle \cong }
 
 Fpn, hence V can be considered as an n-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the field Fp. Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism V 
 
 
 
 
 
 →
 ≅
 
 
 
 
 {\displaystyle {\overset {\cong }{\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-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> 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.

<h2 id="automorphism-group">Automorphism group</h2>
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 <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> 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 <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a>) 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 <a href="/facts/Automorphism/WnyxNFkQ">automorphisms</a> of V we have Aut(V) = { T : V → V | ker T = 0 } = GLn(Fp), the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> of n × n invertible matrices on Fp.
The automorphism group GL(V) = GLn(Fp) acts <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">transitively</a> 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 <a href="/facts/Group_center/gqAg6B8I">center</a>, which is necessarily invariant under all automorphisms, and thus equals all of G.)

<h2 id="a-generalisation-to-higher-orders">A generalisation to higher orders</h2>
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<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> (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.

<h2 id="related-groups">Related groups</h2>
The <a href="/facts/Extra_special_group/WsCbtnK0">extra special groups</a> are extensions of elementary abelian groups by a cyclic group of order p, and are analogous to the <a href="/facts/Heisenberg_group/rsqq1426">Heisenberg group</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Elementary_group/Ijq1Ap03">Elementary group</a></li>
<li><a href="/facts/Hamming_space/220rxsQJ">Hamming space</a></li></ul>

<h2 id="references">References</h2>

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

Elementary abelian group open-in-new

Elementary abelian group