Normal Subgroups and Isomorphism Theorems

Welcome to this comprehensive guide on Normal Subgroups and Isomorphism Theorems, two fundamental concepts in abstract algebra that unlock deeper insights into group structures. Understanding normal subgroups is crucial for constructing quotient groups, which in turn form the basis for the powerful isomorphism theorems. These theorems provide elegant ways to relate different groups, often simplifying complex structures into more familiar forms. This article will break down each concept, illustrate with examples, and connect them into a coherent framework.

Normal Subgroups: The Foundation for Quotient Groups

In group theory, not all subgroups are created equal. While regular subgroups H <= G have left and right cosets, aH and Ha, these are generally not the same. This disparity poses a problem when we want to form a new group from these cosets. For coset multiplication (Ha)(Hb) = H(ab) to be well-defined, we need cosets to respect this operation, which implies aH = Ha for all a in G. This is precisely where normal subgroups come into play.

Defining Normal Subgroups and Their Properties

A subgroup H of a group G is called a normal subgroup of G, written H E G, if ghg^-1 is an element of H for all g in G and all h in H. A key equivalence is that H E G if and only if gH = Hg for all g in G (meaning left and right cosets coincide).

Special cases include the trivial normal subgroups: {e} E G and G E G are always true. Furthermore, every subgroup of an abelian group is normal.

Three Equivalent Conditions for Normality (Theorem 5.1): Let H <= G. The following are equivalent:

1. H E G
2. gHg^-1 is a subset of H for all g in G
3. gHg^-1 = H for all g in G

The Index-2 Theorem: A Quick Test for Normality

If H <= G and the index [G : H] = 2, then H E G. This theorem provides a useful shortcut. If a subgroup H divides the group G into exactly two left cosets and two right cosets, it must be normal. For example, the alternating group A_n is a normal subgroup of the symmetric group S_n because [S_n : A_n] = 2.

Example from S3: In S_3, let H = <(123)> = {e, (123), (132)}. Since [S_3 : H] = 2, H E S_3. However, for K = {e, (12)}, we find (123)K = {(123), (13)} while K(123) = {(123), (23)}. Since these are not equal, K is not a normal subgroup of S_3.

The Kernel: A Special Normal Subgroup

One of the most powerful tools for identifying normal subgroups is the kernel of a homomorphism. The kernel of any homomorphism is always a normal subgroup of the domain. If psi: G -> H is a homomorphism, its kernel, ker(psi) = {g in G | psi(g) = 1_H}, is always normal in G. For example, the kernel of the sign map psi: S_3 -> Z_2 is A_3, which we already know is normal in S_3.

Quotient Groups: