Lattice theorem
From Wikipedia, the free encyclopedia
In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that there exists a bijection from the set of all subgroups of a group G that contain a normal subgroup N onto the set of all subgroups of the quotient group G / N. This means that 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 an antitone 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, for a group G and a normal subgroup N of G, there exists a bijection from the set of all subgroups A of G containing N onto the set of subgroups A' of G / N that maps a subgroup A of G to a subgroup A' = A / N of G / N. For all 
containing N, and subgroups of G / N A' = A / N and B' = B / N, the following hold:
if and only if 
- if , then the index of A in B equals the index 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.
[edit] See also
[edit] References
- W.R. Scott: Group Theory, Prentice Hall, 1964.
