# Correspondence theorem (group theory)

(Redirected from Lattice theorem)

In the area of mathematics known as group theory, the correspondence theorem,[1][2][3][4][5][6][7][8] sometimes referred to as the fourth isomorphism theorem[6][9][note 1][note 2] or the lattice theorem,[10] states that if ${\displaystyle N}$ is a normal subgroup of a group ${\displaystyle G}$, then there exists a bijection from the set of all subgroups ${\displaystyle A}$ of ${\displaystyle G}$ containing ${\displaystyle N}$, onto the set of all subgroups of the quotient group ${\displaystyle G/N}$. The structure of the subgroups of ${\displaystyle G/N}$ is exactly the same as the structure of the subgroups of ${\displaystyle G}$ containing ${\displaystyle N}$, with ${\displaystyle N}$ collapsed to the identity element.

Specifically, if

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

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

${\displaystyle \phi (A)=A/N}$ for all ${\displaystyle A\in {\mathcal {G}}.}$

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

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

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.

More generally, there is a monotone Galois connection ${\displaystyle (f^{*},f_{*})}$ between the lattice of subgroups of ${\displaystyle G}$ (not necessarily containing ${\displaystyle N}$) and the lattice of subgroups of ${\displaystyle G/N}$: the lower adjoint of a subgroup ${\displaystyle H}$ of ${\displaystyle G}$ is given by ${\displaystyle f^{*}(H)=HN/N}$ and the upper adjoint of a subgroup ${\displaystyle K/N}$ of ${\displaystyle G/N}$ is a given by ${\displaystyle f_{*}(K/N)=K}$. The associated closure operator on subgroups of ${\displaystyle G}$ is ${\displaystyle {\bar {H}}=HN}$; the associated kernel operator on subgroups of ${\displaystyle G/N}$ is the identity.

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

## Notes

1. ^ Some authors use "fourth isomorphism theorem" to designate the Zassenhaus lemma; see for example by Alperin & Bell (p. 13) or Robert Wilson (2009). The Finite Simple Groups. Springer. p. 7. ISBN 978-1-84800-988-2.
2. ^ Depending how one counts the isomorphism theorems, the correspondence theorem can also be called the 3rd isomorphism theorem; see for instance H.E. Rose (2009), p. 78.

## References

1. ^ Derek John Scott Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. p. 64. ISBN 978-3-11-017544-8.
2. ^ J. F. Humphreys (1996). A Course in Group Theory. Oxford University Press. p. 65. ISBN 978-0-19-853459-4.
3. ^ H.E. Rose (2009). A Course on Finite Groups. Springer. p. 78. ISBN 978-1-84882-889-6.
4. ^ J.L. Alperin; Rowen B. Bell (1995). Groups and Representations. Springer. p. 11. ISBN 978-1-4612-0799-3.
5. ^ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 35. ISBN 978-0-8218-4799-2.
6. ^ a b Joseph Rotman (1995). An Introduction to the Theory of Groups (4th ed.). Springer. pp. 37–38. ISBN 978-1-4612-4176-8.
7. ^ W. Keith Nicholson (2012). Introduction to Abstract Algebra (4th ed.). John Wiley & Sons. p. 352. ISBN 978-1-118-31173-8.
8. ^ Steven Roman (2011). Fundamentals of Group Theory: An Advanced Approach. Springer Science & Business Media. pp. 113–115. ISBN 978-0-8176-8301-6.
9. ^ Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013). Abstract Algebra: An Inquiry Based Approach. CRC Press. p. 425. ISBN 978-1-4665-6708-5.
10. ^ W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.