Correspondence theorem (group theory)

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 $N$ is a normal subgroup of a group $G$, then there exists a bijection from the set of all subgroups $A$ of $G$ containing $N$, 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

$\phi(A)=A/N$ 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 $G/N$.

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.

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

See also

• Modular lattice

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. 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.