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单词 SubgroupsOfFiniteCyclicGroup
释义

subgroups of finite cyclic group


Let n be the order of a finite cyclic groupMathworldPlanetmath G.  For every positive divisor (http://planetmath.org/Divisibility) m of n, there exists one and only one subgroupMathworldPlanetmathPlanetmath of order m of G.  The group G has no other subgroups.

Proof.  If g is a generatorPlanetmathPlanetmathPlanetmath of G and  n=mk,  then gk generates the subgroup gk, the order of which is equal to the order of gk, i.e. equal to m.  Any subgroup H of G is cyclic (see http://planetmath.org/node/4097this entry).  If  |H|=m,  then H must have a generator of order m; thus apparently  H=g±k=gk.

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