---
title: The Isomorphism Theorems
module: Subgroups and Quotients
moduleNumber: 3
lessonNumber: 5
order: 305
summary: >
  Four theorems relate homomorphisms, quotients, and subgroup lattices. The
  first identifies the image of a homomorphism with the quotient by its kernel;
  the second and third compute quotients built from two subgroups and quotients
  of quotients; the fourth matches the subgroups of $G/N$ with the subgroups of
  $G$ lying above $N$. Together they make quotient groups computable.
topics: [Subgroups and Quotients]
sources:
  - book: Dummit & Foote
    ref: "Ch. 3; §3.3 The Isomorphism Theorems"
  - book: Judson
    ref: "Ch. 11 Homomorphisms; §11.1 Group Homomorphisms, §11.2 The Isomorphism Theorems"
draft: false
---

A [homomorphism](/abstract-algebra/groups-and-symmetry/homomorphisms-and-group-actions)
compresses a group onto its image, and its
[kernel](/abstract-algebra/subgroups-and-quotients/cosets-lagrange-and-normal-subgroups)
records what gets crushed to the identity. The isomorphism theorems make that
compression exact: the image is a copy of the quotient by the kernel, and every
quotient arises this way. Three further theorems handle quotients assembled from
two subgroups, quotients of quotients, and the correspondence of subgroup
lattices.

## The first isomorphism theorem

> **Theorem (First Isomorphism Theorem).** If $\varphi : G \to H$ is a
> homomorphism, then $\ker \varphi \trianglelefteq G$ and
> $$
> G / \ker \varphi \;\cong\; \varphi(G).
> $$

The kernel is normal because it is a kernel. The isomorphism sends the coset
$g \ker\varphi$ to $\varphi(g)$; this is well defined and injective precisely
because two elements share a coset exactly when they have the same image, and it
is surjective onto $\varphi(G)$ by construction.[^df-first] Equivalently, a
homomorphism factors into a projection followed by an isomorphism.

$$
% caption: Every homomorphism factors as the projection onto the quotient by its
% kernel, followed by an isomorphism onto its image inside the codomain.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, minimum width=16mm, minimum height=9mm, font=\small}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[nd] (G) at (0,1.7) {$G$};
  \node[nd] (H) at (5.0,1.7) {$H$};
  \node[nd] (Q) at (0,-0.4) {$\frac{G}{\ker f}$};
  \node[nd, draw=acc] (im) at (5.0,-0.4) {$\mathrm{im}\, f$};
  \draw[->, thick] (G) -- (H) node[midway, above, font=\scriptsize]{homomorphism};
  \draw[->, thick] (G) -- (Q) node[midway, left, font=\scriptsize]{project};
  \draw[->, acc, thick] (Q) -- (im) node[midway, below, font=\scriptsize]{induced iso};
  \draw[->, black] (im) -- (H) node[midway, right, font=\scriptsize]{include};
\end{tikzpicture}
$$

> **Corollary.** For a homomorphism $\varphi : G \to H$: (1) $\varphi$ is
> injective if and only if $\ker \varphi = 1$; (2)
> $[G : \ker \varphi] = |\varphi(G)|$.

The second part is a counting law: the number of cosets of the kernel equals the
size of the image. For linear maps of vector spaces it reappears as
$\dim V = \operatorname{rank}\varphi + \operatorname{nullity}\varphi$, the
rank-nullity theorem.[^df-cor]

> **Worked example.** The special linear group as a kernel. The map
> $\det : \mathrm{GL}_n(F) \to F^\times$ is a surjective homomorphism (every
> nonzero scalar $\lambda$ is the determinant of
> $\operatorname{diag}(\lambda, 1, \dots, 1)$), and its kernel is by definition
> $\mathrm{SL}_n(F)$, the matrices of determinant $1$. The first isomorphism
> theorem gives, in one line,
> $$
> \mathrm{SL}_n(F) \trianglelefteq \mathrm{GL}_n(F),
> \qquad
> \mathrm{GL}_n(F) / \mathrm{SL}_n(F) \;\cong\; F^\times .
> $$
> No coset argument is needed: normality and the isomorphism type of the
> quotient both come from exhibiting the right homomorphism. Over a finite field
> with $q$ elements this also counts:
> $[\mathrm{GL}_n(\mathbb{F}_q) : \mathrm{SL}_n(\mathbb{F}_q)] = q - 1$.[^df-gl]

> **Worked example.** A quotient of $\mathbb{Z}/24\mathbb{Z}$. Take
> $G = \mathbb{Z}/24\mathbb{Z}$ and
> $N = \langle \bar{12} \rangle = \{\bar{0}, \bar{12}\}$. The reduction map
> $\varphi : \mathbb{Z}/24\mathbb{Z} \to \mathbb{Z}/12\mathbb{Z}$,
> $\varphi(\bar{a}) = a \bmod 12$, is a well-defined surjective homomorphism
> whose kernel is exactly $N$, so
> $(\mathbb{Z}/24\mathbb{Z}) / \langle \bar{12} \rangle \cong \mathbb{Z}/12\mathbb{Z}$,
> matching the naive cancellation of $24/2 = 12$.[^df-z24] The same template
> identifies any quotient of a cyclic group without inspecting a single coset
> table.

The other three theorems are consequences of the first applied to well-chosen
homomorphisms.

## The second (diamond) isomorphism theorem

> **Theorem (Second Isomorphism Theorem).** Let $A, B \le G$ with
> $A \le N_G(B)$. Then $AB \le G$, $B \trianglelefteq AB$,
> $A \cap B \trianglelefteq A$, and
> $$
> AB / B \;\cong\; A / (A \cap B).
> $$

Since $A$ normalizes $B$, the product $AB$ is a subgroup and $B$ is normal in
it. Define $\varphi : A \to AB/B$ by $\varphi(a) = aB$; it is a surjective
homomorphism with kernel $\{a \in A \mid aB = B\} = A \cap B$, and the first
theorem finishes it.[^df-second] The name comes from the shape of the four
subgroups in the lattice: $AB$ on top, $A$ and $B$ on the sides, $A \cap B$ at
the bottom, with the two slanted edges carrying isomorphic quotients.

$$
% caption: The diamond of the second isomorphism theorem, with $D = A \cap B$;
% opposite edges carry isomorphic quotients, so $AB/B \cong A/(A \cap B)$.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=3pt, fill=acc!8, font=\small}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[nd, draw=acc] (top) at (0,2.4) {$AB$};
  \node[nd] (A) at (-1.9,0.9) {$A$};
  \node[nd] (B) at (1.9,0.9)  {$B$};
  \node[nd, draw=acc] (bot) at (0,-0.6) {$D$};
  \draw[black] (A) -- (top) node[midway, above left, font=\scriptsize]{iso};
  \draw[black] (B) -- (top);
  \draw[black] (bot) -- (A);
  \draw[black] (bot) -- (B) node[midway, below right, font=\scriptsize]{iso};
  \node[font=\scriptsize, black, anchor=west] at (2.3,1.75) {$\frac{AB}{B}$};
  \node[font=\scriptsize, black, anchor=east] at (-2.3,0.15) {$\frac{A}{D}$};
\end{tikzpicture}
$$

This recovers the product-order formula
$|AB| = |A|\,|B| / |A \cap B|$ in the special case $A \le N_G(B)$, now as an
isomorphism rather than a count. Two cautions attach
to the picture. The unmarked edges of the diamond are not isomorphic to each
other in general: $AB/A$ need not even be a group, since $A$ need not be normal
in $AB$; only the index equation $[AB : A] = [B : A \cap B]$ survives on that
side. And the hypothesis $A \le N_G(B)$ is automatic whenever
$B \trianglelefteq G$, which is the usual way the theorem is invoked: for any
subgroup $A$ and any normal subgroup $B$, the conclusion holds with no further
checking.

> **Worked example.** The diamond in $\mathbb{Z}$. Everything is normal, since
> the group is abelian; take $A = 4\mathbb{Z}$ and $B = 6\mathbb{Z}$. Then
> $A + B = \gcd(4,6)\mathbb{Z} = 2\mathbb{Z}$ and
> $A \cap B = \operatorname{lcm}(4,6)\mathbb{Z} = 12\mathbb{Z}$, so the diamond
> reads
> $$
> \frac{2\mathbb{Z}}{6\mathbb{Z}} \;\cong\; \frac{4\mathbb{Z}}{12\mathbb{Z}},
> $$
> and both sides are cyclic of order $3$. The diamond turns a statement about
> gcd and lcm into a single lattice picture.

## The third isomorphism theorem

> **Theorem (Third Isomorphism Theorem).** Let $H, K \trianglelefteq G$ with
> $H \le K$. Then $K/H \trianglelefteq G/H$ and
> $$
> (G/H) \big/ (K/H) \;\cong\; G/K.
> $$

Define $\varphi : G/H \to G/K$ by $gH \mapsto gK$; because $H \le K$, this is
well defined, and it is a surjective homomorphism with kernel $K/H$.[^df-third]
The mnemonic is **invert and cancel**, as with fractions: the two $H$'s cancel.
The theorem says taking a quotient of a quotient yields no structure that a
single quotient does not already give.

For a numeric instance, take $G = \mathbb{Z}/24\mathbb{Z}$ with
$H = \langle \bar{12} \rangle$ and $K = \langle \bar{6} \rangle$, so
$H \le K$ and both are normal (the group is abelian). Then
$G/H \cong \mathbb{Z}/12\mathbb{Z}$ from the first-theorem example above,
$K/H$ is the order-$2$ subgroup $\langle \bar{6} \rangle / \langle \bar{12} \rangle$
inside it, and the theorem promises
$(G/H)/(K/H) \cong G/K \cong \mathbb{Z}/6\mathbb{Z}$ — the fraction
$\tfrac{24/12}{6/12}$ canceling to $\tfrac{24}{6}$.

$$
% caption: A quotient of a quotient collapses back down: dividing $G/H$ by $K/H$
% cancels the $H$ and lands on $G/K$.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=3pt, fill=acc!8, font=\small}]
  \definecolor{acc}{HTML}{4A6FA5}
  % original chain
  \node[nd] (G) at (0,2.6) {$G$};
  \node[nd] (K) at (0,1.3) {$K$};
  \node[nd] (H) at (0,0.0) {$H$};
  \node[nd] (one) at (0,-1.3) {$1$};
  \draw[black] (H) -- (K) -- (G); \draw[black] (one) -- (H);
  \node[font=\scriptsize, black, anchor=west] at (0.4,-0.65) {$\frac{K}{H}$};
  % arrow
  \draw[->, acc, thick] (1.7,0.65) -- (3.3,0.65) node[midway, above, font=\scriptsize]{mod $H$};
  % quotient chain
  \node[nd] (Gq) at (5.0,2.0) {$\frac{G}{H}$};
  \node[nd] (Kq) at (5.0,0.65) {$\frac{K}{H}$};
  \node[nd] (oneq) at (5.0,-0.7) {$1$};
  \draw[black] (Kq) -- (Gq); \draw[black] (oneq) -- (Kq);
  \node[nd, draw=acc, anchor=west] at (6.5,0.65) {iso $\frac{G}{K}$};
\end{tikzpicture}
$$

## The fourth (lattice) isomorphism theorem

The last theorem is the reason the quotient's lattice can be read off the
[lattice](/abstract-algebra/subgroups-and-quotients/generation-and-subgroup-lattices)
of $G$: collapse $N$ to a point, and everything above $N$ survives, faithfully.

> **Theorem (Fourth / Lattice Isomorphism Theorem).** Let
> $N \trianglelefteq G$. There is a bijection
> $$
> \{\, A \le G \mid N \le A \,\} \;\longleftrightarrow\; \{\, \overline{A} \le G/N \,\},
> \qquad A \mapsto \overline{A} = A/N.
> $$
> For $A, B \ge N$ it satisfies: (1) $A \le B \iff \overline{A} \le \overline{B}$;
> (2) $[B : A] = [\overline{B} : \overline{A}]$; (3)
> $\overline{\langle A, B \rangle} = \langle \overline{A}, \overline{B} \rangle$;
> (4) $\overline{A \cap B} = \overline{A} \cap \overline{B}$; (5)
> $A \trianglelefteq G \iff \overline{A} \trianglelefteq G/N$.

The correspondence sends each subgroup above $N$ to its image, and each subgroup
of $G/N$ back to its complete preimage under the natural projection.[^df-fourth]
It preserves the entire lattice architecture: containment, index, joins, meets,
and normality. The lattice of $G/N$ coincides with the top of the lattice of $G$,
the part sitting above $N$.

> **Worked example.** $Q_8$ modulo its center. The center of the
> [quaternion group](/abstract-algebra/groups-and-symmetry/matrix-and-quaternion-groups)
> is $Z = \{\pm 1\}$, and the subgroups of $Q_8$ containing $Z$ are $Z$ itself,
> the three cyclic subgroups $\langle i \rangle, \langle j \rangle, \langle k \rangle$
> of order $4$, and $Q_8$. The theorem says $Q_8/Z$ has exactly this lattice
> with $Z$ collapsed to the identity: a group of order $4$ with three subgroups
> of order $2$. That forces $Q_8/Z \cong V_4$, the Klein four-group, read off the
> lattice alone, with no multiplication of cosets. Subgroups of $Q_8$ **not**
> containing $Z$ would be invisible in the quotient, but $Q_8$ has none besides
> $1$; in general (e.g. $D_8$ with its five order-$2$ subgroups, only one
> containing the center) several subgroups of $G$ project onto the same subgroup
> of $G/N$, and only the ones above $N$ are counted by the bijection.

$$
% caption: The lattice correspondence: subgroups of $G$ containing $N$ match
% subgroups of $G/N$, preserving containment and index; $N$ maps to the trivial
% subgroup.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=2.5pt, fill=acc!8, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  % left lattice: subgroups of G above N
  \node[nd] (G) at (0,3.0) {$G$};
  \node[nd] (A) at (-1.0,1.6) {$A$};
  \node[nd] (B) at (1.0,1.6)  {$B$};
  \node[nd, draw=acc] (N) at (0,0.2) {$N$};
  \draw[black] (A) -- (G); \draw[black] (B) -- (G);
  \draw[black] (N) -- (A); \draw[black] (N) -- (B);
  % right lattice: subgroups of G/N
  \begin{scope}[xshift=6.0cm]
    \node[nd] (Gq) at (0,3.0) {$\frac{G}{N}$};
    \node[nd] (Aq) at (-1.0,1.6) {$\frac{A}{N}$};
    \node[nd] (Bq) at (1.0,1.6)  {$\frac{B}{N}$};
    \node[nd, draw=acc] (Nq) at (0,0.2) {$1$};
    \draw[black] (Aq) -- (Gq); \draw[black] (Bq) -- (Gq);
    \draw[black] (Nq) -- (Aq); \draw[black] (Nq) -- (Bq);
  \end{scope}
  \draw[->, acc, thick] (1.9,1.6) -- (4.1,1.6) node[midway, above, font=\scriptsize]{mod $N$};
\end{tikzpicture}
$$

## Defining maps on quotients

The theorems share a mechanism: to build a homomorphism out of $G/N$, build one
out of $G$ and check that it sends $N$ to $1$. A homomorphism
$\Phi : G \to H$ **factors through** $N$, inducing a well-defined
$\varphi : G/N \to H$ with $\varphi(gN) = \Phi(g)$, if and only if
$N \le \ker \Phi$.[^df-factor] This single criterion is what makes each
isomorphism above well defined.

Quotient information also flows back down to $G$. A standard specimen:

> **Proposition.** If $G/Z(G)$ is cyclic, then $G$ is abelian (and so
> $G = Z(G)$: the quotient $G/Z(G)$ is never a nontrivial cyclic group).

If $G/Z(G) = \langle xZ(G) \rangle$, every coset is $x^a Z(G)$, so every element
of $G$ is $x^a z$ with $z$ central. Two such elements commute:

$$
(x^a z_1)(x^b z_2) = x^{a+b} z_1 z_2 = (x^b z_2)(x^a z_1),
$$

since powers of $x$ commute with each other and $z_1, z_2$ commute with
everything.[^df-gz] The proposition is a template for the whole method: prove
something about the small group $G/N$, and pull the conclusion back through the
projection. Its immediate payoff is that any group of order $p^2$ (prime $p$)
is abelian — the center is nontrivial by a counting argument met in the
[class equation](/abstract-algebra/group-actions-and-sylow/conjugation-and-the-class-equation),
so $G/Z(G)$ has order $1$ or $p$, both cyclic.

| Theorem | Setup | Statement |
| --- | --- | --- |
| First | $\varphi : G \to H$ | $G/\ker\varphi \cong \operatorname{im}\varphi$ |
| Second | $A \le N_G(B)$ | $AB/B \cong A/(A \cap B)$ |
| Third | $H \le K$, both $\trianglelefteq G$ | $(G/H)/(K/H) \cong G/K$ |
| Fourth | $N \trianglelefteq G$ | subgroups above $N$ $\leftrightarrow$ subgroups of $G/N$ |

With these theorems, a quotient can be identified with a familiar group through
its image rather than handled as an abstract set of cosets.

[^df-first]: **Dummit & Foote**, _Abstract Algebra_, §3.3, Theorem 16 — the First Isomorphism Theorem, $G/\ker\varphi \cong \varphi(G)$.
[^df-cor]: **Dummit & Foote**, §3.3, Corollary 17 — injectivity via trivial kernel, and $[G:\ker\varphi] = |\varphi(G)|$, with the rank-nullity analogy.
[^df-gl]: **Dummit & Foote**, §3.1, Exercise 35 and §3.3, Exercise 1 — $\mathrm{SL}_n(F) \trianglelefteq \mathrm{GL}_n(F)$ with quotient $F^\times$, and the index $q - 1$ over a finite field of order $q$.
[^df-z24]: **Dummit & Foote**, §3.1, Exercise 20 — $(\mathbb{Z}/24\mathbb{Z})/\langle \bar{12} \rangle \cong \mathbb{Z}/12\mathbb{Z}$.
[^df-second]: **Dummit & Foote**, §3.3, Theorem 18 — the Second (Diamond) Isomorphism Theorem.
[^df-third]: **Dummit & Foote**, §3.3, Theorem 19 — the Third Isomorphism Theorem and the "invert and cancel" mnemonic.
[^df-fourth]: **Dummit & Foote**, §3.3, Theorem 20 — the Fourth (Lattice) Isomorphism Theorem and its five preservation properties.
[^df-factor]: **Dummit & Foote**, §3.3 — a homomorphism on $G$ induces one on $G/N$ exactly when $N \le \ker\Phi$ (factoring through $N$).
[^df-gz]: **Dummit & Foote**, §3.1, Exercise 36 — if $G/Z(G)$ is cyclic then $G$ is abelian.