Math 2.0

Group theory and generalizations Concept

Lattice of subgroups

Builds on SubgroupSubgroup generated by a subsetIntersection of subgroups is a subgroup

Summary

The subgroups of a group GG, ordered by inclusion, form a complete lattice. The meet of a family of subgroups is their intersection, and the join is the subgroup generated by their union. The least element is the trivial subgroup {e}\{e\} and the greatest element is GG.

Remarks

For two subgroups HH and KK, the meet is H∩KH \cap K and the join is ⟨H∪K⟩\langle H \cup K \rangle. The lattice of subgroups of (ℤ,+)(\mathbb{Z}, +) is the set of the subgroups nℤn\mathbb{Z}, and mℤ⊆nℤm\mathbb{Z} \subseteq n\mathbb{Z} exactly when nn divides mm. The lattice of normal subgroups of a group is a modular lattice.

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Group, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionLattice of subgroups, concept, no evidence levelLattice of subgroupsconcept
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  1. Group concept
  2. Subgroup definition
  3. Subgroup criterion statement
  4. Intersection of subgroups is a subgroup statement
  5. Subgroup generated by a subset definition
  6. Lattice of subgroups concept

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_of_subgroups, lead section
  2. Web https://en.wikipedia.org/wiki/Lattice_of_subgroups, Properties
About this page
Tag0B8BW4YWTF
URIm2://0B8BW4YWTF
Aliasessubgroup-lattice
Kindconcept
WritingW1 stated
MSC20E15
UsesSubgroup, Subgroup generated by a subset, Intersection of subgroups is a subgroup
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