Correspondence theorem

In group theory, the correspondence theorem^[1]^[2]^[3]^[4]^[5]^[6]^[7]^[8] (also the lattice theorem,^[9] and variously and ambiguously the third and fourth isomorphism theorem^[6]^[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\triangleleft G

, a normal subgroup of G,

{\mathcal {G}}=\{A\mid N\subseteq A<G\}

, the set of all subgroups A of G that contain N, and

{\mathcal {N}}=\{S\mid S<G/N\}

, 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 ${\mathcal {G}}$ then

$A\subseteq B$ if and only if $A/N\subseteq B/N$ ;
if $A\subseteq B$ then $|B:A|=|B/N:A/N|$ , where $|B:A|$ is the index of A in B (the number of cosets bA of A in B);
$\langle A,B\rangle /N=\left\langle A/N,B/N\right\rangle ,$ where $\langle A,B\rangle$ is the subgroup of $G$ generated by $A\cup B;$
$(A\cap B)/N=A/N\cap B/N$ , and
$A$ is a normal subgroup of $G$ if and only if $A/N$ 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. A proof of the correspondence theorem can be found here.

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

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.
^ ^a ^b 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.
^ W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.
^ Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013). Abstract Algebra: An Inquiry Based Approach. CRC Press. p. 425. ISBN 978-1-4665-6708-5.

[Robinson2003-1] Derek John Scott Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. p. 64. ISBN 978-3-11-017544-8.

[Humphreys1996-2] J. F. Humphreys (1996). A Course in Group Theory. Oxford University Press. p. 65. ISBN 978-0-19-853459-4.

[Rose2009-3] H.E. Rose (2009). A Course on Finite Groups. Springer. p. 78. ISBN 978-1-84882-889-6.

[AlperinBell1995-4] J.L. Alperin; Rowen B. Bell (1995). Groups and Representations. Springer. p. 11. ISBN 978-1-4612-0799-3.

[Isaacs1994-5] I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 35. ISBN 978-0-8218-4799-2.

[Rotman1995-6] Joseph Rotman (1995). An Introduction to the Theory of Groups (4th ed.). Springer. pp. 37–38. ISBN 978-1-4612-4176-8.

[Nicholson2012-7] W. Keith Nicholson (2012). Introduction to Abstract Algebra (4th ed.). John Wiley & Sons. p. 352. ISBN 978-1-118-31173-8.

[Roman2011-8] Steven Roman (2011). Fundamentals of Group Theory: An Advanced Approach. Springer Science & Business Media. pp. 113–115. ISBN 978-0-8176-8301-6.

[9] W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.

[HodgeSchlicker2013-10] Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013). Abstract Algebra: An Inquiry Based Approach. CRC Press. p. 425. ISBN 978-1-4665-6708-5.

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Correspondence theorem

See also

References

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