Review of Group Theory: The Fourth Isomorphism Theorem (The Correspondence Theorem)
Point of post: In this post we describe the Fourth Isomorphism Theorem. We’ll then define simple groups and maximal subgroups and prove the fundamental result that is maximal if and only if is simple.
Intuitively the Fourth Isomorphism Theorem says that if and are groups and then there is an inclusion preservering bijective correspondence between the subgroups of contaning and the subgroups of
The Fourth Isomorphism Theorem
We cut right to the chase:
Theorem (Fourth Isomorphism Theorem): Let and be groups and (this means that is an epimorphism). Then, if and then
is an order preserving bijection. Also, if we have that if and only if and if this condition is satisfied then .
Proof: To prove that is injective suppose that and . We note then that it must be true that but note that since we have that
from where it follows that and thus is injective. Now, to see that is surjective we let . Note then that and and thus is surjective, and so consequently bijective. To see that is inclusion preserving we merely note that then .
To prove the second part of the theorem we merely note that if then (since is an epimorphism and we’ve already proven this). Conversely, suppose that . Then, since using the same techniques above we know that and thus (since the preimage of normal subgroups are normal under homomorphisms as we’ve already proven). Now, assuming to prove that we define
since is surjective clearly is surjective. Moreover, we see that since we may use the techniques above to show that . Thus,
Thus, by the First Isomorphism Theorem we may conclude that
the conclusion follows.
From this we have the following nice corollary:
Corollary: Let be a group and . Then, for any there exists a unique such that . Moreover, if and only if .
Proof: Consider the canonical projection
By the Fourth Isomorphism Theorem we have that the mapping
is a bijection, consequently for some . But, . The second part follows immediately from the second part of the Fourth Isomorphism Theorem.
Simple Groups and Maximal Subgroups
A group is called simple if implies that or . In other words, a group is simple if it has no non-trivial normal subgroups. For a group a proper normal subgroup is called maximal if implies or . We now use the Fourth Isomorphism Theorem to show how the two are realted:
Theorem: Let be a group and . Then, is maximal if and only if is simple.
Proof: This an immediate consequence of the last corollary to the Fourth Isomorphism Theorem. In particular, first assume that is maximal. Then, if we have that for some but since is maximal this implies that so that or as required.
Conversely, suppose that is simple and . Then, and so or . But, by assumption this can only be true if or from where the conclusion follows.
1. Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.
2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200