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In group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be 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 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/''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 general, a (possibly infinite) elementary abelian ''p''-group is a direct sum 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. == Examples and properties == * The elementary abelian group (Z/2Z)2 has four elements: . Addition is performed componentwise, taking the result mod 2. For instance, . 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''-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. :: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Elementary abelian group」の詳細全文を読む スポンサード リンク
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