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.

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.

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

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

Notes

↑ 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.
↑ 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

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

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Correspondence theorem (group theory)

See also

Notes

References