Normal Subgroup
A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group: \(H\) is normal if and only if \(gHg^{-1} = H\) for any \(g \in G.\) Equivalently, a subgroup \(H\) of \(G\) is normal if and only if \(gH = Hg\) for any \(g \in G\).
Normal subgroups are useful in constructing quotient groups, and in analyzing homomorphisms.
Quotient Groups
Main wiki: Quotient groups
A quotient group is defined as \(G/N\) for some normal subgroup \(N\) of \(G\), which is the set of cosets of \(N\) w.r.t. \(G\), equipped with the operation \(\circ\) satisfying \[(gN) \circ (hN) = (gh)N\] for all \(g,h \in G\).
This definition is the reason that \(N\) must be normal to define a quotient group; it holds because the chain of equalities
\[(gN)(hN) = g(Nh)N = g(hN)N = (gh)(NN) = (gh)N\]
holds, where \(g(Nh)N = g(hN)N\) utilizes the fact that \(Nh = hN\) for any \(h\) (true iff \(N\) is normal, by definition).
For example, consider the subgroup \(H = \{0, 2, 4, 6\}\) of \(G=\mathbb{Z}_8\) (which is an additive group). The left cosets are
\[\{0 + h | h \in H\} = \{2 + h | h \in H\} = \{4 + h | h \in H\} = \{6 + h | h \in H\} = \{0, 2, 4, 6\}\] \[\{1 + h | h \in H\} = \{3 + h | h \in H\} = \{5 + h | h \in H\} = \{7 + h | h \in H\} = \{1, 3, 5, 7\},\]
so \(G/H = \big\{\{0, 2, 4, 6\}, \{1, 3, 5, 7\}\big\}\). This can be more cleanly written as \[G/H = \{0 + H, 1 + H\},\] which is isomorphic to \(\{0,1\}\), or the cyclic group \(C_2\). Additional examples:
- The quotient group \(\mathbb{Z}/2\mathbb{Z}\), where \(2\mathbb{Z}\)--the group of even integers--is a normal subgroup of \(\mathbb{Z}\), is isomorphic to \(C_2\) as well.
- The quotient group \(\mathbb{R}/\mathbb{Z}\), where \(\mathbb{Z}\)--the group of integers--is a normal subgroup of the reals \(\mathbb{R}\), is isomorphic to the circle group defined by the complex numbers with magnitude 1.
Homomorphisms and Normal Subgroups
Recall that a homomorphism from \(G\) to \(H\) is a function \(\phi\) such that
\[\phi(g_1g_2) = \phi(g_1)\phi(g_2)\]
for all \(g_1, g_2 \in G\). The kernel of a homomorphism is the set of elements of \(G\) that are sent to the identity in \(H\), and the kernel of any homomorphism is necessarily a normal subgroup of \(G\).
In fact, more is true: the image of \(G\) under this homomorphism (the set of elements \(G\) is sent to under \(\phi\)) is isomorphic to the quotient group \(G/\text{ker}(\phi)\), by the first isomorphism theorem. This provides a bijection between normal subgroups of \(G\) and the set of images of \(G\) under homomorphisms.
Thus normal subgroups can be classified in another manner:
A subgroup \(N\) of \(G\) is normal if and only if there exists a homomorphism on \(G\) whose kernel is \(N\).