- Lattice theorem
-
In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups of the quotient group G / N. The structure of the subgroups of G / N is exactly the same as the structure of the subgroups of G containing N, with N collapsed to the identity element.
This establishes a monotone Galois connection between the lattice of subgroups of G and the lattice of subgroups of G / N, where the associated closure operator on subgroups of G 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
- ϕ(A) = A / N for all
One further has that if A and B are in
, and A' = A/N and B' = B/N, then
if and only if
;
- if
then B:A = B':A', where B:A is the index of A in B (the number of cosets bA of A in B);
where
is the subgroup of G generated by
and
- A is a normal subgroup of G if and only if A' is a normal subgroup of G'.
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.
See also
References
- W.R. Scott: Group Theory, Prentice Hall, 1964.
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