Normal subgroups; quotient groups

At a Glance

• Frequency: 2 sub-parts across 2 of 13 years (2014, 2019)
• Priority tier: T3
• Marks (count): 10 (2)
• Average solve time: ~7 min
• Difficulty mix: medium 2
• Section: A | Dominant type: proof / enumeration

Why This Chapter Matters

Normal subgroup questions appear in two flavours: direct conjugation tests (2014, 10 marks) and listing all quotient groups of a finite abelian group (2019, 10 marks). The conjugation test requires explicit matrix calculation and a clean statement of the normality criterion $gNg^{-1}\subseteq N$. The quotient-listing question reduces to enumerating divisors and applying the bijection between subgroups of a cyclic group and divisors of its order. Both are reliable marks-earners that can be set up in 2 minutes if you know the normal subgroup equivalences.

Minimum Theory

Normal subgroup. A subgroup $N\le G$ is normal (written $N\trianglelefteq G$) if any of the following equivalent conditions holds: (i) $gNg^{-1}=N$ for all $g\in G$; (ii) left and right cosets coincide, $gN=Ng$ for all $g\in G$; (iii) $N$ is the kernel of some group homomorphism from $G$.

Quotient group. If $N\trianglelefteq G$, the set of cosets $G/N=\{gN:g\in G\}$ forms a group under $(g_1N)(g_2N)=g_1g_2N$, with $|G/N|=|G|/|N|$ by Lagrange.

Subgroups of $\mathbb Z_n$. For each divisor $d\mid n$ there is exactly one subgroup of order $d$, namely $\langle n/d\rangle$. Every subgroup is normal (abelian group). The quotient $\mathbb Z_n/\langle n/d\rangle$ is cyclic of order $d$: $\mathbb Z_n/\langle n/d\rangle \cong \mathbb Z_d$.

Question Archetypes

normal-subgroup-test (1 question(s); 2014)

Recognition Cues — a subgroup defined by a structural constraint (e.g., matrices with specified diagonal entries); asked to show it is normal, often after first verifying the ambient set is a group.

Solution Template

1. (If needed) Verify $G$ is a group: check closure, note associativity is inherited, find identity, compute inverses.
2. Write a general element $n\in N$ and a general element $g\in G$.
3. Compute $g\,n\,g^{-1}$ explicitly.
4. Show the result satisfies the defining property of $N$ — it lies in $N$.
5. Conclude $gNg^{-1}\subseteq N$ for all $g\in G$; hence $N\trianglelefteq G$.

Worked Example

2014 Paper 2, 2014-P2-Q1a (10 marks)

Let $G$ be the group of all real $2\times 2$ matrices $\begin{bmatrix}x & y\\ 0 & z\end{bmatrix}$ with $xz\ne0$, under matrix multiplication. Let $N = \left\{\begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix} : a\in\mathbb R\right\}$. Is $N$ a normal subgroup of $G$?

Part 1 — $G$ is a group. Write $A=\begin{pmatrix}x_1&y_1\\0&z_1\end{pmatrix}$, $B=\begin{pmatrix}x_2&y_2\\0&z_2\end{pmatrix}$.

Closure: $AB=\begin{pmatrix}x_1x_2 & x_1y_2+y_1z_2\\ 0 & z_1z_2\end{pmatrix}$. Upper-triangular; product of diagonal entries is $(x_1z_1)(x_2z_2)\ne0$. $\checkmark$

Associativity: inherited from matrix multiplication. $\checkmark$

Identity: $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ has $1\cdot1=1\ne0$, so $I\in G$. $\checkmark$

Inverses: $A^{-1}=\begin{pmatrix}1/x & -y/(xz)\\ 0 & 1/z\end{pmatrix}\in G$ since $(1/x)(1/z)=1/(xz)\ne0$. $\checkmark$

Hence $G$ is a group.

Part 2 — Normality. Take $n_a=\begin{pmatrix}1&a\\0&1\end{pmatrix}\in N$ and general $g=\begin{pmatrix}x&y\\0&z\end{pmatrix}\in G$:

$g\,n_a = \begin{pmatrix}x&xa+y\\0&z\end{pmatrix},$

$g\,n_a\,g^{-1} = \begin{pmatrix}x&xa+y\\0&z\end{pmatrix}\begin{pmatrix}1/x & -y/(xz)\\0&1/z\end{pmatrix} = \begin{pmatrix}1 & xa/z\\ 0 & 1\end{pmatrix} = n_{xa/z} \in N.$

Since $gNg^{-1}\subseteq N$ for every $g\in G$:

$\boxed{N \text{ is a normal subgroup of } G.}\;\blacksquare$

(Alternative: $N=\ker\phi$ where $\phi(g)=(x,z)$ maps $G\to\mathbb R^*\times\mathbb R^*$ — kernels are always normal.)

Common Traps

• Computing $gn_ag^{-1}$ only for specific values of $g$ does not prove normality; the calculation must work for a general $g$.
• The off-diagonal entry after conjugation is $xa/z$, not $a$ itself. The key is that $xa/z$ is still real, so $n_{xa/z}\in N$ for every $a\in\mathbb R$.
• In Part 1, the $(1,2)$ entry of $AB$ is $x_1y_2+y_1z_2$, not $y_1+y_2$ — write it correctly.

list-quotients (1 question(s); 2019)

Recognition Cues — “Write down all quotient groups of $\mathbb Z_n$”; any finite abelian cyclic group; asked for all (normal) subgroups and the corresponding quotients.

Solution Template

1. Observe $\mathbb Z_n$ is abelian, so every subgroup is normal.
2. List all divisors $d$ of $n$.
3. For each $d$, the unique subgroup of order $d$ is $H_d=\langle n/d\rangle$.
4. The quotient $\mathbb Z_n/H_d$ has order $n/d$; since a quotient of a cyclic group is cyclic, $\mathbb Z_n/H_d\cong\mathbb Z_{n/d}$.
5. Include the two improper quotients: $d=1$ gives $\mathbb Z_n/\{0\}\cong\mathbb Z_n$; $d=n$ gives $\mathbb Z_n/\mathbb Z_n\cong\{0\}$.

Worked Example

2019 Paper 2, 2019-P2-Q2b (10 marks)

Write down all quotient groups of the group $\mathbb Z_{12}$.

$\mathbb Z_{12}$ is abelian, so every subgroup is normal. Divisors of $12$: $1,2,3,4,6,12$.

| Subgroup $H_d$ | $|H_d|=d$ | Generator | $|\mathbb Z_{12}/H_d|$ | Quotient | |---|---|---|---|---| | $\{0\}$ | $1$ | — | $12$ | $\mathbb Z_{12}$ | | $\{0,6\}$ | $2$ | $6$ | $6$ | $\mathbb Z_6$ | | $\{0,4,8\}$ | $3$ | $4$ | $4$ | $\mathbb Z_4$ | | $\{0,3,6,9\}$ | $4$ | $3$ | $3$ | $\mathbb Z_3$ | | $\{0,2,4,6,8,10\}$ | $6$ | $2$ | $2$ | $\mathbb Z_2$ | | $\mathbb Z_{12}$ | $12$ | $1$ | $1$ | $\{0\}$ |

$\boxed{\mathbb Z_{12}/H \;\cong\; \mathbb Z_{12},\;\mathbb Z_6,\;\mathbb Z_4,\;\mathbb Z_3,\;\mathbb Z_2,\;\{0\}.}$

There are exactly 6 quotient groups, one for each of the 6 divisors of 12. $\blacksquare$

Common Traps

• “All quotient groups” includes the improper ones: $G/\{0\}\cong G$ and $G/G\cong\{0\}$. Missing either loses marks.
• For a cyclic group all quotients are cyclic — there is no $\mathbb Z_2\times\mathbb Z_2$ quotient of $\mathbb Z_{12}$.
• A quick check: $|H_d|\cdot|\mathbb Z_{12}/H_d|$ must equal $12$ in every row (Lagrange).

Marks-Aware Writing

Both questions are 10 marks. For the normal subgroup test: state the normality criterion $gNg^{-1}\subseteq N$ explicitly, carry out the full computation for a general $g$, and box the conclusion. Doing only specific examples scores at most 4 marks. For the quotient-group enumeration: present a complete table with subgroup, order, and quotient name; verify the count equals the number of divisors; state that all quotients are cyclic. Missing the two improper quotients loses 2–3 marks.

Practice Set

• 2018-P2-Q2a (15 m) — — Hint: define $\phi:\mathbb R\to S^1$ by $\phi(x)=e^{2\pi ix}$, compute the kernel, apply the First Isomorphism Theorem.