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单词 CharacterizationOfFiniteNilpotentGroups
释义

characterization of finite nilpotent groups


Let G be a finite groupMathworldPlanetmath. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    G is nilpotentPlanetmathPlanetmathPlanetmath.

  2. 2.

    Every subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) of G is subnormal.

  3. 3.

    Every proper subgroupMathworldPlanetmath of G is properly contained in its normalizerMathworldPlanetmath.

  4. 4.

    Every maximal subgroup of G is normal.

  5. 5.

    Every Sylow subgroup of G is normal.

  6. 6.

    G is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups) of p-groups (http://planetmath.org/PGroup4).

For proofs, see the article on finite nilpotent groups.

Condition 3 above is the normalizer condition.

Titlecharacterization of finite nilpotent groups
Canonical nameCharacterizationOfFiniteNilpotentGroups
Date of creation2013-03-22 13:16:24
Last modified on2013-03-22 13:16:24
Owneryark (2760)
Last modified byyark (2760)
Numerical id11
Authoryark (2760)
Entry typeTheorem
Classificationmsc 20D15
Classificationmsc 20F18
Related topicFiniteNilpotentGroups
Related topicNilpotentGroup
Related topicNormalizerCondition
Related topicSubnormalSubgroup
Related topicLocallyNilpotentGroup
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