---
title: Generation and the Lattice of Subgroups
module: Subgroups and Quotients
moduleNumber: 3
lessonNumber: 3
order: 303
summary: >
  The subgroup generated by a subset $A$ is the smallest subgroup containing it,
  described top-down as an intersection and bottom-up as the set of words in $A$
  and its inverses. Collecting all subgroups and ordering them by containment
  produces the subgroup lattice, whose Hasse diagram shows the joins, meets, and
  containment relations among all subgroups.
topics: [Subgroups and Quotients]
sources:
  - book: Dummit & Foote
    ref: "Ch. 2; §2.4 Subgroups Generated by Subsets of a Group"
  - book: Dummit & Foote
    ref: "§2.5 The Lattice of Subgroups of a Group"
draft: false
---

Taking all powers of a single element $x$ builds the
[cyclic subgroup](/abstract-algebra/subgroups-and-quotients/cyclic-groups)
$\langle x \rangle$. Replacing that one element with an arbitrary subset $A$
gives the general construction, the subgroup generated by $A$. It has two
descriptions, one existential and one computational, and they agree.

## The subgroup generated by a set

The top-down description rests on one closure fact.

> **Proposition.** The intersection of any nonempty collection of subgroups of
> $G$ is a subgroup of $G$.

This is the subgroup criterion applied to the intersection: the identity lies in
every member so in the intersection, and if $a, b$ lie in every member then so
does $ab^{-1}$.[^df-int] Intersecting all subgroups that contain $A$ therefore
produces a subgroup, and it is the smallest one containing $A$.

> **Definition (Generated subgroup).** For $A \subseteq G$, the **subgroup
> generated by $A$** is
> $$
> \langle A \rangle = \bigcap_{\substack{A \subseteq H \\ H \le G}} H,
> $$
> the intersection of all subgroups of $G$ containing $A$. It is the unique
> minimal subgroup containing $A$.

For a finite set we write $\langle a_1, \dots, a_n \rangle$, and for subsets
$A, B$ we write $\langle A, B \rangle$ for $\langle A \cup B \rangle$. This
description proves existence and uniqueness but says nothing about the elements.
The bottom-up description supplies them: close $A$ under the operation and
inverses.

> **Proposition (Words).** Let $\overline{A}$ be the set of all finite products
> $a_1^{\varepsilon_1} a_2^{\varepsilon_2} \cdots a_k^{\varepsilon_k}$ with
> $a_i \in A$ and $\varepsilon_i = \pm 1$ (the empty product being $1$). Then
> $\overline{A} = \langle A \rangle$.

Such a product is a **word** in $A$. The set $\overline{A}$ is a subgroup, since
$(a_1^{\varepsilon_1} \cdots a_k^{\varepsilon_k})(b_1^{\delta_1} \cdots b_m^{\delta_m})^{-1}$
is again a word; it contains $A$; and any subgroup containing $A$ must contain
every word, so it contains $\overline{A}$.[^df-words] The two descriptions
bracket $\langle A \rangle$ from above and below and meet.

$$
% caption: A word in the generators $a$ and $b$: a finite string of the
% generators and their inverses, multiplied left to right, is a typical element
% of $\langle a, b \rangle$.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  tok/.style={draw, minimum size=8mm, font=\small}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[tok, fill=acc!10, draw=acc] (t1) at (0,0)   {$a$};
  \node[tok] (t2) at (1.0,0) {$b$};
  \node[tok, fill=acc!10, draw=acc] (t3) at (2.0,0) {$a$};
  \node[tok] (t4) at (3.0,0) {$b$};
  \node[tok, fill=acc!10, draw=acc] (t5) at (4.0,0) {$a$};
  \node[font=\scriptsize, black, above] at (t2.north) {inverse};
  \node[font=\scriptsize, black, above] at (t4.north) {inverse};
  \draw[->, acc, thick] (-0.7,-0.75) -- (4.7,-0.75)
    node[midway, below, font=\scriptsize, black]{multiply left to right};
\end{tikzpicture}
$$

## Abelian and non-abelian generation

When $G$ is abelian the generators commute, so every word collects its factors
by generator: $\langle a_1, \dots, a_k \rangle = \{ a_1^{n_1} \cdots a_k^{n_k} \mid n_i \in \mathbb{Z} \}$.
If each $a_i$ has finite order $d_i$, the group has at most $d_1 d_2 \cdots d_k$
elements. Orders of generators bound the order of the group.

This fails in a non-abelian group. Take $D_8 = \langle r, s \rangle$
and set $a = s$, $b = rs$. Both have order $2$, and since $r = (rs)s = ba$ lies
in $\langle a, b \rangle$, the pair generates all of $D_8$. Two elements of order
$2$ generate a group of order $8$; the word $aba$ cannot be shortened to
$a^\alpha b^\beta$. More sharply, for every $n$ the group
$D_{2n} = \langle a, b \rangle$ with $|a| = |b| = 2$ has order $2n$, so long
alternating words $abab\cdots$ never collapse.[^df-nonab] Two further examples:

- $S_n = \langle (1\,2), (1\,2\,3\cdots n) \rangle$ is generated by an element of
  order $2$ and one of order $n$, yet $|S_n| = n!$.
- In $\mathrm{GL}_2(\mathbb{R})$, the matrices
  $a = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and
  $b = \begin{pmatrix} 0 & 2 \\ 1/2 & 0 \end{pmatrix}$ satisfy $a^2 = b^2 = I$,
  yet $ab = \begin{pmatrix} 1/2 & 0 \\ 0 & 2 \end{pmatrix}$ has infinite order
  (its powers are $\operatorname{diag}(2^{-n}, 2^{n})$), so
  $\langle a, b \rangle$ is an **infinite** subgroup generated by two elements
  of order $2$.

For a non-abelian $G$ and a random subset $A$, even the order of
$\langle A \rangle$ is generally out of reach. Useful generation comes from
chosen subsets: if $y$ commutes with $x$, or normalizes $\langle x \rangle$,
then $\langle x, y \rangle$ stays controlled — abelian in the first case, with
order bounded by $|x|\,|y|$.

> **Worked example.** Two transpositions generate $S_3$. Take
> $A = \{(1\,2), (2\,3)\}$ in $S_3$. The words of length two already produce
> $(1\,2)(2\,3) = (1\,2\,3)$ and $(2\,3)(1\,2) = (1\,3\,2)$, and length three
> gives $(1\,2)(2\,3)(1\,2) = (1\,3)$. With the identity that is all six
> elements, so $\langle (1\,2), (2\,3) \rangle = S_3$. The count agrees with
> Lagrange run backwards: the subgroup contains elements of order $2$ and $3$,
> so its order is divisible by $6$, forcing the whole group. Any two
> **distinct** transpositions in $S_3$ generate it the same way.

## The lattice of subgroups

Collect every subgroup of a finite $G$ and order them by containment. The result
is drawn as a **Hasse diagram**: plot subgroups with larger ones higher, and
draw an edge upward from $A$ to $B$ when $A < B$ with no subgroup strictly
between. This graph is the **lattice of subgroups**, and it displays the
structure of $G$ better than its multiplication table.[^df-lat]

Two operations read off the diagram. For subgroups $H$ and $K$:

- **Join.** The smallest subgroup containing both, $\langle H, K \rangle$, found
  by tracing upward from $H$ and $K$ to their lowest common ancestor.
- **Meet.** The largest subgroup contained in both, $H \cap K$ (a subgroup by
  the intersection proposition), found by tracing downward.

$$
% caption: The join and meet of two subgroups: $\langle H, K \rangle$ is their
% lowest common upper bound and $H \cap K$ their highest common lower bound in
% the lattice.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=2.5pt, fill=acc!8, font=\small}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[nd, draw=acc] (top) at (0,2.4) {$< H$, $K >$};
  \node[nd] (h) at (-1.7,0.8) {$H$};
  \node[nd] (k) at (1.7,0.8) {$K$};
  \node[nd, draw=acc] (bot) at (0,-0.8) {$H$ meet $K$};
  \draw[black] (h) -- (top);
  \draw[black] (k) -- (top);
  \draw[black] (bot) -- (h);
  \draw[black] (bot) -- (k);
  \node[acc, font=\scriptsize, anchor=west] at (1.0,2.4) {join};
  \node[acc, font=\scriptsize, anchor=west] at (1.4,-0.8) {meet};
\end{tikzpicture}
$$

The dihedral group $D_8$ has ten subgroups, and its lattice is the standard
reference picture. Between the trivial subgroup and $D_8$ sit five subgroups of
order $2$ and three of order $4$; the center $\langle r^2 \rangle$ is the unique
order-$2$ subgroup lying inside all three order-$4$ subgroups.

$$
% caption: The subgroup lattice of $D_8$; three subgroups of order four, five of
% order two, with the center $\langle r^2 \rangle$ shared by every order-four
% subgroup.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=2.2pt, fill=acc!8, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[nd, draw=acc] (G) at (0,3.4) {$D_8$};
  \node[nd] (V1) at (-3.0,1.9) {$< s$, $r^2 >$};
  \node[nd] (R)  at (0,1.9)    {$< r >$};
  \node[nd] (V2) at (3.0,1.9)  {$< sr$, $r^2 >$};
  \node[nd] (s)   at (-4.0,0.4) {$< s >$};
  \node[nd] (sr2) at (-2.2,0.4) {$< sr^2 >$};
  \node[nd, draw=acc] (c) at (0,0.4) {$< r^2 >$};
  \node[nd] (sr)  at (2.2,0.4)  {$< sr >$};
  \node[nd] (sr3) at (4.0,0.4)  {$< sr^3 >$};
  \node[nd] (one) at (0,-1.1)   {$1$};
  \draw[black] (V1) -- (G); \draw[black] (R) -- (G); \draw[black] (V2) -- (G);
  \draw[black] (s) -- (V1); \draw[black] (sr2) -- (V1); \draw[black] (c) -- (V1);
  \draw[black] (c) -- (R);
  \draw[black] (c) -- (V2); \draw[black] (sr) -- (V2); \draw[black] (sr3) -- (V2);
  \draw[black] (one) -- (s); \draw[black] (one) -- (sr2); \draw[black] (one) -- (c);
  \draw[black] (one) -- (sr); \draw[black] (one) -- (sr3);
\end{tikzpicture}
$$

The [quaternion group](/abstract-algebra/groups-and-symmetry/matrix-and-quaternion-groups)
$Q_8$ contrasts sharply. It has the same order as $D_8$ and also three
subgroups of order $4$, but a single subgroup of order $2$: its center, shared
by all three. Its lattice is slimmer, and the difference between it and the
divisor lattice of a cyclic group of order $8$ measures the gap between
a non-abelian and a cyclic group.

$$
% caption: The subgroup lattice of $Q_8$; the three order-four subgroups
% $\langle i \rangle, \langle j \rangle, \langle k \rangle$ all meet in the
% single order-two center.
\begin{tikzpicture}[>=stealth, font=\footnotesize,
  nd/.style={draw, inner sep=2.5pt, fill=acc!8, font=\scriptsize}]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[nd, draw=acc] (G) at (0,3.0) {$Q_8$};
  \node[nd] (i) at (-2.2,1.6) {$< i >$};
  \node[nd] (j) at (0,1.6)    {$< j >$};
  \node[nd] (k) at (2.2,1.6)  {$< k >$};
  \node[nd, draw=acc] (z) at (0,0.2) {$Z(Q_8)$};
  \node[nd] (one) at (0,-1.2) {$1$};
  \draw[black] (i) -- (G); \draw[black] (j) -- (G); \draw[black] (k) -- (G);
  \draw[black] (z) -- (i); \draw[black] (z) -- (j); \draw[black] (z) -- (k);
  \draw[black] (one) -- (z);
\end{tikzpicture}
$$

## Reading structure off the diagram

Beyond joins and meets, the lattice computes centralizers and normalizers with
little arithmetic. In $D_8$, to find $C_{D_8}(s)$: first $r^2$ commutes with
$s$, so $\langle s, r^2 \rangle \le C_{D_8}(s)$. The only subgroups above
$\langle s, r^2 \rangle$ are itself and $D_8$, and $C_{D_8}(s) \neq D_8$ because
$r$ does not commute with $s$. The lattice leaves one option:
$C_{D_8}(s) = \langle s, r^2 \rangle$.[^df-read]

Two cautions keep the picture honest. Isomorphic groups have identical lattices,
but the converse fails: nonisomorphic groups can share a lattice ($\mathbb{Z}_2 \times \mathbb{Z}_8$
and the modular group of order $16$ have the same diagram), so the lattice is a
partial invariant, not a fingerprint. And for larger groups the lattice need
not be planar; the diagram of $D_{16}$ cannot be drawn without crossings.

For infinite groups the full lattice cannot be drawn at all, but **partial
lattices** still carry arguments: draw only the subgroups in play, with an edge
meaning containment rather than "nothing strictly between." Most lattice
reasoning in later chapters, the isomorphism theorems especially, happens in
such fragments.

The integers are the model infinite case. Every subgroup of $\mathbb{Z}$ is
cyclic, $\langle m \rangle = m\mathbb{Z}$ for a unique $m \ge 0$, and
containment is divisibility reversed: $m\mathbb{Z} \le d\mathbb{Z}$ if and only
if $d \mid m$. Joins and meets become gcd and lcm,

$$
\langle m\mathbb{Z}, n\mathbb{Z} \rangle = \gcd(m,n)\,\mathbb{Z},
\qquad
m\mathbb{Z} \cap n\mathbb{Z} = \operatorname{lcm}(m,n)\,\mathbb{Z},
$$

so the subgroup lattice of $\mathbb{Z}$ is the divisibility lattice of the
positive integers turned upside down, with $\mathbb{Z} = 1\mathbb{Z}$ on top and
no bottom other than the trivial subgroup, which is the meet of every infinite
chain $2\mathbb{Z} > 4\mathbb{Z} > 8\mathbb{Z} > \cdots$. Any finite fragment of
this lattice is a fragment of number theory.

## Finitely generated groups and maximal subgroups

Generation gives a size vocabulary for infinite groups.

- **Finitely generated.** $G$ is **finitely generated** if $G = \langle A \rangle$
  for some finite $A$. Every finite group qualifies (take $A = G$), and
  $\mathbb{Z} = \langle 1 \rangle$ shows an infinite group can be generated by
  one element.
- **Not finitely generated.** $\mathbb{Q}$ under addition is not finitely
  generated: any finite set of fractions has a common denominator $k$, so it
  generates only a subgroup of $\langle 1/k \rangle \cong \mathbb{Z}$, never all
  of $\mathbb{Q}$. In fact every finitely generated subgroup of $\mathbb{Q}$ is
  cyclic.[^df-fg]
- **Maximal subgroup.** $M < G$ is **maximal** if no subgroup sits strictly
  between $M$ and $G$: in the Hasse diagram, the maximal subgroups sit
  one edge below the top. In a finite group every proper subgroup lies
  inside a maximal one. In $D_{2n}$ the rotation subgroup $\langle r \rangle$ is
  maximal (index $2$); in a cyclic group of order $n$ the maximal subgroups are
  the $\langle x^p \rangle$ for primes $p \mid n$, matching the lattice of
  divisors for [cyclic groups](/abstract-algebra/subgroups-and-quotients/cyclic-groups).

Even so, the lattice is the first thing to sketch when meeting a new finite
group.

[^df-int]: **Dummit & Foote**, _Abstract Algebra_, §2.4, Proposition 8 — the intersection of a nonempty collection of subgroups is a subgroup.
[^df-words]: **Dummit & Foote**, §2.4, Proposition 9 — $\langle A \rangle$ equals the set of finite words in $A$ and its inverses.
[^df-nonab]: **Dummit & Foote**, §2.4 — the $D_{2n} = \langle a, b \rangle$, $S_n$, and $\mathrm{GL}_2(\mathbb{R})$ examples showing generator orders do not bound the order of the generated subgroup.
[^df-lat]: **Dummit & Foote**, §2.5 — the construction of the subgroup lattice, joins and meets, and the lattices of $\mathbb{Z}/n\mathbb{Z}$, $V_4$, $S_3$, $D_8$, $Q_8$, and $D_{16}$.
[^df-read]: **Dummit & Foote**, §2.5 — computing centralizers and normalizers from the subgroup lattice, illustrated with $C_{D_8}(s)$; partial lattices for infinite groups; the modular group of order 16 sharing the lattice of $\mathbb{Z}_2 \times \mathbb{Z}_8$.
[^df-fg]: **Dummit & Foote**, §2.4, Exercises 14–16 — finitely generated groups, every finitely generated subgroup of $\mathbb{Q}$ is cyclic, $\mathbb{Q}$ is not finitely generated, and maximal subgroups (rotations in $D_{2n}$, $\langle x^p \rangle$ in a cyclic group).