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单词 GeneralizedDihedralGroup
释义

generalized dihedral group


Let A be an abelian groupMathworldPlanetmath.The generalized dihedral group Dih(A)is the semidirect productMathworldPlanetmath AC2,where C2 is the cyclic groupMathworldPlanetmath of order 2,and the generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Generator) of C2 maps elements of A to their inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

If A is cyclic, then Dih(A) is called a dihedral groupMathworldPlanetmath.The finite dihedral group Dih(Cn) is commonly denoted by Dn or D2n(the differing conventions being a source of confusion).The infinite dihedral group Dih(C) is denoted by D,and is isomorphicPlanetmathPlanetmathPlanetmath tothe free productMathworldPlanetmath C2*C2 of two cyclic groups of order 2.

If A is an elementary abelian 2-group, then so is Dih(A).If A is not an elementary abelian 2-group, then Dih(A) is non-abelianMathworldPlanetmathPlanetmath.

The subgroupMathworldPlanetmathPlanetmath A×{1} of Dih(A) is of index 2,and every element of Dih(A) that is not in this subgroup has order 2.This property in fact characterizes generalized dihedral groups,in the sense that if a group G has a subgroup N of index 2 such that all elements of the complement GN are of order 2,then N is abelian and GDih(N).

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更新时间:2025/5/4 1:32:56