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.

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.

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 $N$.

• Subgroup
• Homomorphism
• Isomorphism