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Let \(\phi: G\to G'\) be a group homomorphism. If \(N\) is a normal subgroup of \(G\), then \(\phi(N)\) is a normal subgroup of \(\phi(G)\). Conversely, if \(N'\) is a normal subgroup of \(\phi(G)\), then \(\phi^{-1}(N')\) is a normal subgroup of \(G\).
注意到 scope 的不同！第一項 \(\phi(N)\) is a normal subgroup of \(\phi(G)\)，僅是 \(\phi(G)\) 的 subgroup 而非整個 \(G'\)；第二項則是整個 \(G\)。
Let \(G\) be a group. Then the subgroup
\[Z(G) = \{x\in G\vert xg = gx, \forall g\in G\}\]is called the center of \(G\).
Center 的大小標示著 \(G\) 的可交換性。
\(Z(G)\) is a normal subgroup of \(G\).
prove that \(Z(G)\) is normal
\[gZ(G) = \{gx\vert x \in Z(G) \} = \{xg \vert x\in Z(G)\} = Z(G)g. \tag*{$\blacksquare$}\]Let \(G\) be a group. An element of the form \(a^{-1}b^{-1}ab\) is a commutator of the group. The subgroup generated by all the commutators is called the commutator subgroup, and is denoted by \([G,G]\).
Commutator 的大小標示著 \(G\) 的不可交換性。
對於 \(S_n\) 來說，commutator 一定是 even permutation！
Prove that \([S_n, S_n] = A_n\).
Suppose that \(N\) is a normal subgroup of \(G\) such that \(G/N\) is abelian. Then we have \((aN)(bN) = (bN)(aN)\), or equivalently,
\[\begin{align*} &abN = baN, \\ &\Rightarrow (ba)^{-1}abN = N, \\ &\Rightarrow a^{-1}b^{-1}ab \in N. \end{align*}\]Thus \(N\) must contain all the commutators.
The definition of commutator subgroups arises naturally when we try to determine when \(G/N\) is abelian.
The commutator subgroup \([G,G]\) is a normal subgroup. Moreover, suppose that \(N\) is a normal subgroup of \(G\). Then \(G/N\) is abelian iff \([G,G]<N\).
Proof
We first show that \([G,G]\) is a normal subgroup of \(G\), i.e. \([G,G]\triangleleft G\). It suffices to prove that the generators (commutators) satisfy \(g^{-1}(a^{-1}b^{-1}ab)g \in [G,G]\) for all \(a,b,g \in G\).
見 remark。
We have
\[\begin{align*} g^{-1}(a^{-1}b^{-1}ab)g &= g^{-1}a^{-1}b^{-1}a(gbb^{-1}g^{-1})bg \\ &= g^{-1}a^{-1}b^{-1}agb(b^{-1}g^{-1}bg) \\ &= [(ag)^{-1}b^{-1}(ag)b](b^{-1}g^{-1}bg), \end{align*}\]which is the product of two commutators. Thus, \(g^{-1}(a^{-1}b^{-1}ab)g\in [G,G]\), and \([G,G] \triangleleft G\).
The proof of the second statement has already been shown in the previous remark. ◼
Center 和 commutator 是一種相反的概念：對於可交換群 \(G\)，\(Z(G) = G\) 而 \([G,G] = \{e\}\)。
以 \(S_3\) 為例。\(S_3\) 共有 \(6\) 個 subgroups：
\[\{e\}, {\langle (123) \rangle}, {\langle (23) \rangle}, {\langle (12) \rangle}, {\langle (13) \rangle}, S_3。\]可以明顯看出，\({\langle (23) \rangle}, {\langle (12) \rangle}, {\langle (13) \rangle}\) 都不 normal（\(gH \not = Hg\), for \(g = (123)\)），所以 \(S_3\) 的 normal subgroups 共有 \(\{e\}, \langle (123) \rangle, S_3\)。
直接從定義看，\(Z(G)\) 只可能是 \(\{e\}\)。因為 \(S_3\) 不是 abelian，所以 \([S_3, S_3] \not = \{e\}\)，又因為 commutator 一定是 even permutation，所以 \([S_3,S_3] \not = S_3\)；於是 \([S_3, S_3] = \langle (123) \rangle\)。
When \(H\) is not normal,
\[\bigcap_{g\in G} gHg^{-1}\]is called the normal core of \(H\), which is the largest normal subgroup contained in \(H\).
改成 \(\bigcup\) 就不是 subgroup 了！
From this theorem, we know that every normal subgroup \(N\) of \(G\) is a kernel of some group homomorphism \(\rho\). To gain more insight of \(N\), we shall look for a more canonical homomorphism.
e.g.
這兩種 homomorphism 的 kernel 都是特殊的 normal subgroup。
第一例中的 \(\rho\) 其實是 homomorphism，可以試著驗證看看；底下證明 \(\text{ker}(\rho) = Z(G)\)。
Proof
For all \(g \in \text{ker}(\rho)\), we have
\[\begin{align*} &\rho(g) = I_G, \text{where } I_G \text{ is the identity automorphism on } G. \\ \Rightarrow\ &\rho_g(x) = I_G(x) = x,\ \forall x \in G \\ \Rightarrow\ &gxg^{-1} = x \\ \Rightarrow\ &gx = xg,\ \forall x \in G. \end{align*}\]Thus, \(g \in Z(G)\). ◼
第二個例子稱作 generalized Caley theorem。\(S_{G/H}\) 相較於 Cayley theorem 的 \(S_G\)，是較小的群，因此比較容易應用，但這是有代價的：Cayley theorem 中的 \(\text{ker}(\phi) = \{e\}\)，而此處 \(\text{ker}(\rho) = \bigcap_{g\in G}gHg^{-1}\)，也就是說，\(\rho\) 不是 one-to-one！
底下證明 \(\text{ker}(\rho) = \bigcap_{g\in G}gHg^{-1}\)：
Proof
For all \(g \in \text{ker}(\rho)\), we have \(\rho(g) = \lambda_e\), i.e.
\[\rho_g(x) = \lambda_e(xH) \iff gxH = xH.\]That is to say
\[x^{-1}gxH = H \iff x^{-1}gx \in H \iff g \in xHx^{-1},\forall x \in H.\]This means that \(\text{ker}(\rho) = \bigcap_{x\in G}xHx^{-1}\). ◼
事實上，Cayley theorem 是 \(H = \{e\}\) 時的特例。
Generalized Caley theorem 的應用待補。
What is an automorphism? An automorphism is an isomorphism with the same domain and codomain.
\(\rho: G\to G\)，如果 \(\rho\) 是 isomorphism，則 \(\rho\) 又被稱為 automorphism。
Given
\[\text{Aut}(G) = \{\rho:G\to G \vert \rho \text{ is an isomorphism}\},\]then \(\text{Aut}(G)\) is called the group of automorphism on \(G\), which is in fact a subgroup of \(S_G\).
The set of all isomorphisms of \(G\).
A group \(G\) is simple if \(\{e\}\) and \(G\) are the only two distinct normal subgroups of \(G\).
e.g.
If \(p\) is a prime, then \(\mathbb{Z}_p\) is a simple group.
For \(n\ge 5\), the alternating group \(A_n\) is a simple group.
A normal series of a finite group \(G\) is a sequence of subgroups \(G_i\) of \(G\) satisfies
\[G_0 = \{e\} \triangleleft G_1 \triangleleft \cdots \triangleleft G_k = G.\]Note that \(G_i\) is only normal in \(G_{i+1}\) instead of \(G\), and \(G_i/G_{i+1}\) is simple for all \(i\).
Every finite simple groups is isomorphic to one of the following groups:
- a member of one of three infinite classes of such, namely:
- \(\mathbb{Z}_p\) for all prime \(p\).
- \(A_n\) for \(n\ge 5\).
- the groups of Lie type. (Certain matrix groups over finite fields.)
- one of \(26\) groups called the sporadic groups.
- the Tits group (sometimes considered a \(27\)th sporadic group.)