Theorem 7.4.1.
Let \(N \leq G\text{.}\) The following are equivalent:
-
\(N \trianglelefteq G\text{;}\)
-
\(gNg^{-1} \subset N\) for all \(g \in G\text{;}\)
-
\(gNg^{-1} = N\) for all \(g \in G\text{,}\) i.e. \(N\) is closed under conjugation.
Section 7.4 Homomorphisms, Normal Subgroups and Quotients
A homomorphism between groups \(G\) and \(H\) is a map \(\phi : G \rightarrow H\) such that
\begin{equation*}
\phi(g_1 g_2) = \phi(g_1)\phi(g_2)
\end{equation*}
for all \(g_1, g_2 \in G\text{.}\) The kernel of \(\phi\) is
\begin{equation*}
\ker \phi = \{ g \in G : \phi(g) = e_H \}\text{.}
\end{equation*}
A subgroup \(N \leq G\) is normal in \(G\text{,}\) written \(N \trianglelefteq G\text{,}\) if \(gN = Ng\) for every \(g \in G\text{;}\) that is, its left and right cosets coincide. For \(g \in G\text{,}\) the element \(gag^{-1}\) is the conjugate of \(a\) by \(g\text{,}\) and \(gNg^{-1} = \{gng^{-1} : n \in N\}\) is a conjugate of the subset \(N\text{.}\)
Proof.
(1) \(\Rightarrow\) (2). Since \(gN = Ng\text{,}\) for \(n \in N\) there is \(n' \in N\) with \(gn = n'g\text{,}\) so \(gng^{-1} = n' \in N\text{.}\)
(2) \(\Rightarrow\) (3). Applying (2) to \(g^{-1}\) gives \(g^{-1}Ng \subset N\text{,}\) i.e. \(N \subset gNg^{-1}\text{;}\) combined with \(gNg^{-1} \subset N\text{,}\) this gives equality.
(3) \(\Rightarrow\) (1). For \(n \in N\text{,}\) \(gng^{-1} = n'\) for some \(n' \in N\text{,}\) so \(gn = n'g \in Ng\text{;}\) hence \(gN \subset Ng\text{,}\) and symmetrically \(Ng \subset gN\text{.}\)
Theorem 7.4.2.
The kernel of a homomorphism \(\phi : G \rightarrow H\) is a normal subgroup of \(G\text{.}\)
Proof.
Let \(K = \ker \phi\text{.}\) By Proposition 7.2.1, \(K \leq G\text{:}\) \(e \in K\text{,}\) and if \(k_1, k_2 \in K\) then \(\phi(k_1 k_2^{-1}) = \phi(k_1) \phi(k_2)^{-1} = e_H\text{,}\) so \(k_1 k_2^{-1} \in K\text{.}\) For \(g \in G\) and \(k \in K\text{,}\)
\begin{equation*}
\phi(g k g^{-1}) = \phi(g) \phi(k) \phi(g)^{-1} = \phi(g) e_H \phi(g)^{-1} = e_H\text{,}
\end{equation*}
so \(gkg^{-1} \in K\text{.}\) Thus \(gKg^{-1} \subset K\) for every \(g \in G\text{,}\) and \(K\) is normal by Theorem 7.4.1.
If \(N \trianglelefteq G\text{,}\) the cosets of \(N\) form a group under \((aN)(bN) = abN\text{,}\) the quotient group \(G/N\text{.}\)
Theorem 7.4.3.
If \(N \trianglelefteq G\text{,}\) then \(G/N\) is a group of order \([G:N]\) under \((aN)(bN) = abN\text{.}\)
Proof.
We must first check that the operation is well-defined, i.e. independent of coset representatives. Suppose \(aN = bN\) and \(cN = dN\text{,}\) so \(a = bn_1\) and \(c = dn_2\) for some \(n_1, n_2 \in N\text{.}\) Since \(N\) is normal, \(n_1 d = d n_1'\) for some \(n_1' \in N\text{,}\) so
\begin{equation*}
acN = bn_1 d n_2 N = b d n_1' n_2 N = bdN\text{.}
\end{equation*}
The identity of \(G/N\) is \(eN = N\text{,}\) and the inverse of \(gN\) is \(g^{-1}N\text{;}\) associativity is inherited from \(G\text{.}\) The order of \(G/N\) is the number of cosets of \(N\text{,}\) which is \([G:N]\text{.}\)
Every normal subgroup arises as a kernel: the map \(\phi : G \rightarrow G/N\) given by \(\phi(g) = gN\text{,}\) the canonical homomorphism, is a homomorphism with kernel \(N\text{,}\) since \(\phi(g_1 g_2) = g_1 g_2 N = (g_1 N)(g_2 N) = \phi(g_1)\phi(g_2)\text{.}\) Combined with Theorem 7.4.2, this shows that normal subgroups and kernels of homomorphisms are the same thing. The precise relationship between a homomorphism and the quotient by its kernel is given by the following fundamental result.
Theorem 7.4.4. First Isomorphism Theorem.
Let \(\psi : G \rightarrow H\) be a group homomorphism with \(K = \ker \psi\text{.}\) Then \(G/K\) is isomorphic to the image \(\psi(G)\text{,}\) via the map \(gK \mapsto \psi(g)\text{.}\)
