Summary of Normal Subgroups, Quotient Groups, and Isomorphism Theorems

Normal Subgroups, Quotient Groups & Isomorphism Theorems

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Introduction

Group theory studies algebraic structures called groups and the relationships between them. This material covers normal subgroups, quotient groups, and the three isomorphism theorems. It explains how quotients let us "collapse" part of a group, why normality is required, and how homomorphisms connect groups via kernels and images.

1. Normal Subgroups

Definition: A subgroup $H$ of a group $G$ is a normal subgroup written $H \trianglelefteq G$ if for all $g \in G$ and $h \in H$ we have $ghg^{-1} \in H$.

  • Equivalent condition: $H \trianglelefteq G$ if and only if $gH = Hg$ for all $g \in G$ (left and right cosets coincide).
  • Trivial normal subgroups: ${e}$ and $G$.
  • Every subgroup of an abelian group is normal.

Three equivalent statements (Theorem)

Let $H\le G$. The following are equivalent:

  1. $H \trianglelefteq G$.
  2. $gHg^{-1} \subseteq H$ for all $g \in G$.
  3. $gHg^{-1} = H$ for all $g \in G$.

Proof sketch: (1) implies (2) by conjugating elements; (2) applied to $g^{-1}$ gives (3); (3) implies (1) because $gH = Hg$.

Useful results

  • Index-2 theorem: If $[G:H]=2$ then $H \trianglelefteq G$ because there are only two cosets, so left and right cosets must match.
  • Kernel of any homomorphism is normal: If $\varphi:G\to K$ is a homomorphism then $\ker\varphi \trianglelefteq G$.
💡 Did you know?Did you know that the alternating group $A_n$ is normal in $S_n$ because $[S_n:A_n]=2$ for $n\ge 2$?

2. Quotient Groups

Let $H \trianglelefteq G$. The set of cosets is $$G/H = {Hg\mid g\in G}.$$ Define multiplication by $$Hg_1\cdot Hg_2 = H(g_1g_2).$$

  • This operation is well-defined exactly because $H$ is normal: if $Hg_1=Hg_1'$ and $Hg_2=Hg_2'$ then $H(g_1g_2)=H(g_1'g_2')$.
  • Theorem: $(G/H,\cdot)$ is a group. Identity is $H$, inverse of $Hg$ is $Hg^{-1}$, associativity comes from $G$.
  • If $G$ is finite, $|G/H| = |G|/|H| = [G:H]$.

Definition (Natural map): For $N\trianglelefteq G$ define $\psi:G\to G/N$ by $\psi(g)=Ng$. This is a surjective homomorphism with $\ker\psi=N$.

Practical examples:

  • $\mathbb{Z}/n\mathbb{Z}$ arises from $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}/n\mathbb{Z}$ (quotient by $n\mathbb{Z}$) and is isomorphic to the familiar cyclic group $\mathbb{Z}_n$.
  • For $G=S_3$ and $N=A_3$, $G/N\cong\mathbb{Z}_2$ since cosets correspond to even and odd permutations.
💡 Did you know?Fun fact: Quotient groups let us "zoom out": all elements of the normal subgroup become the identity in the quotient, so structural details inside that subgroup are ignored.

3. Homomorphisms and the Kernel

Definition: For a homomorphism $\psi:G\to H$, the kernel is $\ker\psi={g\in G\mid\psi(g)=1_H}$.

  • Theorem: $\ker\psi \trianglelefteq G$.
  • $\psi$ is injective if and only if $\ker\psi={1_G}$.

4. First Isomorphism Theorem

Theorem (First Isomorphism): If $\psi:G\to H$ is a surjective homomorphism with kernel $K$, then $$G/K \cong H.$$

Construction of the isomorphism: define $\alpha:G/K\to H$ by $\alpha(Kg)=\psi(g)$. This map is well-defined, a homomorphism, injective and surjective, hence an isomorphism.

Examples:

  • Sign map: $\mathrm{sign}:S_n\to{\pm1}$ with kernel $A_n$ gives $S_n/A_n\cong\mathbb{Z}_2$.
  • Reduction modulo $n$: $\psi:\mathbb{Z}\to\mathbb{Z}_n$, $\ker\psi=n\mathbb{Z}$, so $\mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}_n$.
  • Exponential map: $\psi:(\mathbb{R},+)\to(\mathbb{R}_{>0},\times)$, $\psi(x)=e^x$, kernel ${0}$, giving an isomorphism between additive and multiplicative positive reals.

5. Products of Subgroups and the Second Isomorphism Theorem

Definition: For subgroups $H,K\le G$ define $HK={hk\mid h\in H,k\in K}$.

  • Warning: $HK$ need not be a subgroup in general.
  • If $K\trianglelefteq G$ (or $H\trianglelefteq G$) then $HK$ is a subgroup.

Theorem (Second Isomorphism): If $H,K\le G$ with $K\trianglelefteq G$ then

  1. $HK\le G$.
  2. $H\cap K\trianglelefteq H$.
  3. $$H/(H\cap K)\cong HK/K.$$

Proof idea: define $\psi:H\to HK/K$ by $\psi(h)=Kh$, show it is s

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Isomorphism Theorems

Klíčová slova: Group Theory

Klíčové pojmy: Definition: $H\trianglelefteq G$ iff $ghg^{-1}\in H$ for all $g\in G,h\in H$, Normality equivalent to $gH=Hg$ for all $g\in G$, Kernel $\ker\psi$ of any homomorphism $\psi:G\to H$ is normal in $G$, Quotient $G/H$ defined when $H\trianglelefteq G$, with $Hg_1\cdot Hg_2=H(g_1g_2)$, First Isomorphism Theorem: $G/\ker\psi\cong\mathrm{Im}\,\psi$ for homomorphism $\psi$, Second Isomorphism Theorem: if $K\trianglelefteq G$ then $H/(H\cap K)\cong HK/K$, Counting formula: if $K\trianglelefteq G$ then $|HK|=|H|\,|K|/|H\cap K|$, Third Isomorphism Theorem: if $K\subseteq H$ normal then $(G/K)/(H/K)\cong G/H$, Index-2 subgroups are always normal, Natural map $\psi:G\to G/N$ is surjective with kernel $N$

## Introduction Group theory studies algebraic structures called **groups** and the relationships between them. This material covers normal subgroups, quotient groups, and the three isomorphism theorems. It explains how quotients let us "collapse" part of a group, why normality is required, and how homomorphisms connect groups via kernels and images. ## 1. Normal Subgroups > **Definition:** A subgroup $H$ of a group $G$ is a **normal subgroup** written $H \trianglelefteq G$ if for all $g \in G$ and $h \in H$ we have $ghg^{-1} \in H$. - Equivalent condition: $H \trianglelefteq G$ if and only if $gH = Hg$ for all $g \in G$ (left and right cosets coincide). - Trivial normal subgroups: $\{e\}$ and $G$. - Every subgroup of an abelian group is normal. ### Three equivalent statements (Theorem) Let $H\le G$. The following are equivalent: 1. $H \trianglelefteq G$. 2. $gHg^{-1} \subseteq H$ for all $g \in G$. 3. $gHg^{-1} = H$ for all $g \in G$. Proof sketch: (1) implies (2) by conjugating elements; (2) applied to $g^{-1}$ gives (3); (3) implies (1) because $gH = Hg$. ### Useful results - Index-2 theorem: If $[G:H]=2$ then $H \trianglelefteq G$ because there are only two cosets, so left and right cosets must match. - Kernel of any homomorphism is normal: If $\varphi:G\to K$ is a homomorphism then $\ker\varphi \trianglelefteq G$. Did you know that the alternating group $A_n$ is normal in $S_n$ because $[S_n:A_n]=2$ for $n\ge 2$? ## 2. Quotient Groups Let $H \trianglelefteq G$. The set of cosets is $$G/H = \{Hg\mid g\in G\}.$$ Define multiplication by $$Hg_1\cdot Hg_2 = H(g_1g_2).$$ - This operation is well-defined exactly because $H$ is normal: if $Hg_1=Hg_1'$ and $Hg_2=Hg_2'$ then $H(g_1g_2)=H(g_1'g_2')$. - Theorem: $(G/H,\cdot)$ is a group. Identity is $H$, inverse of $Hg$ is $Hg^{-1}$, associativity comes from $G$. - If $G$ is finite, $|G/H| = |G|/|H| = [G:H]$. > **Definition (Natural map):** For $N\trianglelefteq G$ define $\psi:G\to G/N$ by $\psi(g)=Ng$. This is a surjective homomorphism with $\ker\psi=N$. Practical examples: - $\mathbb{Z}/n\mathbb{Z}$ arises from $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}/n\mathbb{Z}$ (quotient by $n\mathbb{Z}$) and is isomorphic to the familiar cyclic group $\mathbb{Z}_n$. - For $G=S_3$ and $N=A_3$, $G/N\cong\mathbb{Z}_2$ since cosets correspond to even and odd permutations. Fun fact: Quotient groups let us "zoom out": all elements of the normal subgroup become the identity in the quotient, so structural details inside that subgroup are ignored. ## 3. Homomorphisms and the Kernel > **Definition:** For a homomorphism $\psi:G\to H$, the **kernel** is $\ker\psi=\{g\in G\mid\psi(g)=1_H\}$. - Theorem: $\ker\psi \trianglelefteq G$. - $\psi$ is injective if and only if $\ker\psi=\{1_G\}$. ## 4. First Isomorphism Theorem > **Theorem (First Isomorphism):** If $\psi:G\to H$ is a surjective homomorphism with kernel $K$, then $$G/K \cong H.$$ Construction of the isomorphism: define $\alpha:G/K\to H$ by $\alpha(Kg)=\psi(g)$. This map is well-defined, a homomorphism, injective and surjective, hence an isomorphism. Examples: - Sign map: $\mathrm{sign}:S_n\to\{\pm1\}$ with kernel $A_n$ gives $S_n/A_n\cong\mathbb{Z}_2$. - Reduction modulo $n$: $\psi:\mathbb{Z}\to\mathbb{Z}_n$, $\ker\psi=n\mathbb{Z}$, so $\mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}_n$. - Exponential map: $\psi:(\mathbb{R},+)\to(\mathbb{R}_{>0},\times)$, $\psi(x)=e^x$, kernel $\{0\}$, giving an isomorphism between additive and multiplicative positive reals. ## 5. Products of Subgroups and the Second Isomorphism Theorem > **Definition:** For subgroups $H,K\le G$ define $HK=\{hk\mid h\in H,k\in K\}$. - Warning: $HK$ need not be a subgroup in general. - If $K\trianglelefteq G$ (or $H\trianglelefteq G$) then $HK$ is a subgroup. > **Theorem (Second Isomorphism):** If $H,K\le G$ with $K\trianglelefteq G$ then 1. $HK\le G$. 2. $H\cap K\trianglelefteq H$. 3. $$H/(H\cap K)\cong HK/K.$$ Proof idea: define $\psi:H\to HK/K$ by $\psi(h)=Kh$, show it is s