Lattice theorem

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In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if is a normal subgroup of a group , then there exists a bijection from the set of all subgroups of such that contains , onto the set of all subgroups of the quotient group . The structure of the subgroups of is exactly the same as the structure of the subgroups of containing with collapsed to the identity element.

This establishes a monotone Galois connection between the lattice of subgroups of and the lattice of subgroups of , where the associated closure operator on subgroups of is

Specifically, if

G is a group,
N is a normal subgroup of G,
is the set of all subgroups A of G such that , and
is the set of all subgroups of G/N,

then there is a bijective map such that

for all

One further has that if A and B are in , and A' = A/N and B' = B/N, then

This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group.

Similar results hold for rings, modules, vector spaces, and algebras.

See also

References

  • W.R. Scott: Group Theory, Prentice Hall, 1964.

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