Lattice of subgroups

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Definition

The lattice of subgroups is a lattice (that is, a special kind of partially ordered set) whose elements are the subgroups are where the partial order is subgroup inclusion. Important points about this lattice:

  • The meet operation in the lattice is intersection of subgroups
  • The join operation in the lattice is join of subgroups
  • The lattice is bounded, with the upper bound being the improper subgroup (or the whole group) and the lower bound being the trivial subgroup.
  • The lattice is a complete lattice, that is, the meet and join operation can both be performed for infinitely many elements.

Importance

Lattice-theoretic properties of subgroups

Further information: Lattice-theoretic properties of subgroups

Complements in lattices