Lattice theorem

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 \bar H = HN.

Specifically, If

G is a group,
N is a normal subgroup of G,
\mathcal{G} is the set of all subgroups A of G such that N\subseteq A\subseteq G, and
\mathcal{N} is the set of all subgroups of G/N,

then there is a bijective map \phi:\mathcal{G}\to\mathcal{N} such that

ϕ(A) = A / N for all A\in \mathcal{G}.

One further has that if A and B are in \mathcal{G}, and A' = A/N and B' = B/N, then

  • A \subseteq B if and only if A' \subseteq B';
  • if A \subseteq B then B:A = B':A', where B:A is the index of A in B (the number of cosets bA of A in B);
  • \langle A,B\rangle / N = \langle A',B' \rangle, where \langle A,B \rangle is the subgroup of G generated by A\cup B;
  • (A\cap B)/N = A' \cap B', 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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