Generation and the Lattice of Subgroups

Taking all powers of a single element $x$ builds the cyclic subgroup$⟨x⟩$. 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.

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 $a{b}^{-1}$.¹ Intersecting all subgroups that contain $A$ therefore produces a subgroup, and it is the smallest one containing $A$.

For a finite set we write $⟨a_1,\dots ,a_n⟩$, and for subsets $A,B$ we write $⟨A,B⟩$ for $⟨A\cup B⟩$. 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.

Such a product is a word in $A$. The set $\overline{A}$ is a subgroup, since $(a_1^{{\epsilon }_1}\cdots a_k^{{\epsilon }_k})(b_1^{{\delta }_1}\cdots b_m^{{\delta }_m}{)}^{-1}$ is again a word; it contains $A$; and any subgroup containing $A$ must contain every word, so it contains $\overline{A}$.² The two descriptions bracket $⟨A⟩$ from above and below and meet.

Abelian and non-abelian generation

When $G$ is abelian the generators commute, so every word collects its factors by generator: $⟨a_1,\dots ,a_k⟩=\{a_1^{n_1}\cdots a_k^{n_k}\mid n_i\in \mathbb{Z}\}$. 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 $D_8=⟨r,s⟩$ and set $a=s$, $b=rs$. Both have order $2$, and since $r=(rs)s=ba$ lies in $⟨a,b⟩$, 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 $D_{2n}=⟨a,b⟩$ with $|a|=|b|=2$ has order $2n$, so long alternating words $abab\cdots$ never collapse.³ Two further examples:

• $S_n=⟨(12),(123\cdots n)⟩$ 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 $a=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and $b=\left(\begin{array}{cc}0& 2\\ 1/2& 0\end{array}\right)$ satisfy $a^2=b^2=I$, yet $ab=\left(\begin{array}{cc}1/2& 0\\ 0& 2\end{array}\right)$ has infinite order (its powers are $\mathrm{diag}(2^{-n},2^n)$), so $⟨a,b⟩$ 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 $⟨A⟩$ is generally out of reach. Useful generation comes from chosen subsets: if $y$ commutes with $x$, or normalizes $⟨x⟩$, then $⟨x,y⟩$ stays controlled — abelian in the first case, with order bounded by $|x||y|$.

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.⁴

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

• Join. The smallest subgroup containing both, $⟨H,K⟩$, 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.

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 $⟨r^2⟩$ is the unique order-$2$ subgroup lying inside all three order-$4$ subgroups.

The quaternion group$Q_8$ 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.

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 $⟨s,r^2⟩\le C_{D_8}(s)$. The only subgroups above $⟨s,r^2⟩$ are itself and $D_8$, and $C_{D_8}(s)\ne D_8$ because $r$ does not commute with $s$. The lattice leaves one option: $C_{D_8}(s)=⟨s,r^2⟩$.⁵

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, $⟨m⟩=m\mathbb{Z}$ 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,

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 $G=⟨A⟩$ for some finite $A$. Every finite group qualifies (take $A=G$), and $\mathbb{Z}=⟨1⟩$ 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 $⟨1/k⟩\cong \mathbb{Z}$, never all of $\mathbb{Q}$. In fact every finitely generated subgroup of $\mathbb{Q}$ is cyclic.⁶
• 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 $⟨r⟩$ is maximal (index $2$); in a cyclic group of order $n$ the maximal subgroups are the $⟨x^p⟩$ for primes $p\mid n$, matching the lattice of divisors for cyclic groups.

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