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In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for ''n''≥4 the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold. For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4. == Definition and classification == A group homomorphism from ''D'' to ''G'' is said to be a Schur cover of the finite group ''G'' if: # the kernel is contained both in the center and the derived subgroup of ''D'', and # amongst all such homomorphisms, this ''D'' has maximal size. The Schur multiplier of ''G'' is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group ''D'' is often called the Schur cover or Darstellungsgruppe. The Schur covers of the symmetric and alternating groups were classified in . The symmetric group of degree ''n'' ≥ 4 has two isomorphism classes of Schur covers, both of order 2⋅''n''!, and the alternating group of degree ''n'' has one isomorphism class of Schur cover, which has order ''n''! except when ''n'' is 6 or 7, in which case the Schur cover has order 3⋅''n''!. 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「covering groups of the alternating and symmetric groups」の詳細全文を読む スポンサード リンク
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