Classifying Groups of Small Order

Classifying the groups of a fixed order $n$ is the concrete form of the Hölder program: find the simple building blocks, then find how they assemble. Three tools do the work. Sylow's theorem produces normal subgroups; direct and semidirect products glue them together; and presentations name the outcome by generators and relations. Free groups make that last notion precise.

The classification strategy

For most small $n$ the plan is uniform.¹ The order factors as $n=p_1^{a_1}\cdots p_k^{a_k}$, and the goal is to force some Sylow subgroup to be normal, then peel the group apart.

The hard step is forcing a normal subgroup. When the Sylow count $n_p$ is pinned to $1$ by the congruence and divisibility constraints, the Sylow $p$-subgroup is unique, hence normal, and gives the factor $H$. The number of gluings $\phi$ is then small.

The base cases

Three families of orders are settled by earlier results, and they cover most small $n$.

• Order $p$. Every group of prime order is cyclic, ${\mathbb{Z}}_p$, by Lagrange: a nonidentity element has order dividing $p$, hence order $p$, so it generates.
• Order $p^2$. Every group of order $p^2$ is abelian, since a $p$-group has nontrivial center and $G/Z(G)$ cyclic forces $G$ abelian. The two abelian types are ${\mathbb{Z}}_{p^2}$ and ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$.
• Order $pq$ ($p<q$ primes). The Sylow $q$-subgroup is normal, so $G\cong {\mathbb{Z}}_q⋊{\mathbb{Z}}_p$. If $p\nmid q-1$ the only group is cyclic ${\mathbb{Z}}_{pq}$; if $p\mid q-1$ there is one additional non-abelian group.

Order $p^3$ adds the first non-abelian $p$-group examples. For $p$ odd there are exactly two non-abelian groups of order $p^3$, distinguished by whether they contain an element of order $p^2$; for $p=2$ these are $D_8$ and $Q_8$. The abelian ones follow the partition count: three of them, from the partitions $3$; $2,1$; $1,1,1$.

Forcing non-simplicity

When no Sylow count is immediately $1$, the classification depends on ruling out simple groups of order $n$ — showing every such group has a proper nontrivial normal subgroup. Three counting techniques recur.²

Counting elements of prime order. If a Sylow $p$-subgroup has order $p$, then distinct conjugates intersect trivially, so the $n_p$ conjugates contribute $n_p(p-1)$ elements of order $p$. Summing these counts over several primes can exceed $|G|$, which is impossible, forcing some $n_p=1$.

Exploiting subgroups of small index. A subgroup $H$ of index $k$ gives an action of $G$ on the $k$ cosets, hence a homomorphism $G\to S_k$ whose kernel lies in $H$. If $G$ is simple the kernel is trivial, so $G$ embeds in $S_k$ and $|G|\mid k!$. This bounds how small an index a simple group can have and often contradicts a Sylow count.

Permutation representations. Refining the previous idea, once $G\le S_k$ one compares Sylow subgroups inside $S_k$. If $P\in {\mathrm{Syl}}_p(G)$ is also Sylow in $S_k$, then $|N_G(P)|$ must divide $|N_{S_k}(P)|$, a numerical constraint that frequently fails.

These methods show, for example, that there are no simple groups of order $56$, $105$, or $396$. Not every order yields to them — genuine simple groups exist at $60$, $168$, $360$, and beyond — but for the small orders below, they always suffice to find a normal subgroup and begin the assembly.

The table of small orders

Combining the base cases with Sylow-driven splitting produces the complete census through order $15$, with order $16$ recorded for scale.³

Prime orders and the order $15$ (where $3\nmid 5-1$) give a single cyclic group. Prime-square orders give two abelian types. Non-abelian groups first appear at order $6$, and the count jumps whenever the order carries a large prime power: order $16$ already has fourteen types, and the $p$-group tables grow rapidly from there.

Free groups

Naming these groups needs a precise notion of generators and relations. A presentation like $D_{2n}=⟨r,s\mid r^n=s^2=1,s^{-1}rs=r^{-1}⟩$ has to mean something definite. The free group supplies the raw material: generators subject to no relations at all.

In $F(\{a,b\})$ the elements are strings like $a$, $ab$, $abab$, $ba$, and $ab{a}^{-1}{b}^2$, all distinct: no relation forces any two reduced words to coincide. Concatenation multiplies them, cancelling only juxtaposed inverses, so $(aba)(a^{-1}b)=a{b}^2$. The Cayley graph of $F(\{a,b\})$ is the infinite $4$-valent tree — no loops, because a loop would be a nontrivial relation.

The defining feature of freeness is a mapping property: because the generators satisfy no relations, they may be sent anywhere.

A homomorphism out of $F(S)$ is fixed by naming the image of each generator, and those images are unconstrained.⁴ This is the exact analogue of a basis of a vector space, where any assignment of the basis vectors extends uniquely to a linear map; $F(S)$ is the vector space with basis $S$ in the category of groups.

Presentations as quotients

Every group is a quotient of a free group. Taking $S$ to be a generating set of $G$, the universal property gives a surjection $\pi :F(S)\to G$, and $G\cong F(S)/\ker \pi$. A presentation records $G$ by the generators $S$ together with enough relations to cut out the kernel.

The kernel is not $R$ itself but the subgroup generated by $R$ and all its conjugates, since the kernel must be normal.⁵ Collapsing that kernel imposes exactly the relations in $R$ and nothing more.

To verify that $⟨S\mid R⟩$ presents a known finite group $G$, two checks suffice: $S$ generates $G$, and any group generated by $S$ subject to $R$ has order at most $|G|$. For $D_{2n}=⟨r,s\mid r^n=s^2=1,s^{-1}rs=r^{-1}⟩$, the relations imply a normal subgroup $⟨r⟩$ of order at most $n$ with quotient generated by $s$ of order at most $2$, so any such group has order at most $2n$; since $D_{2n}$ of order $2n$ satisfies the relations, the presentation is exactly $D_{2n}$.⁵

Familiar groups get compact presentations:

Presentations are compact but hard to decode: in general it is undecidable whether a given set of generators and relations defines the trivial group or an infinite one, so a compact presentation may hide a group that is hard to identify.⁶

The reach and limit of the program

Assembling Sylow's theorem, products, and presentations classifies every group through order $15$ and, with more labor, well beyond. The method is inductive: each group breaks into a normal subgroup and a complement, both smaller and already understood. It stalls at the simple groups, which admit no such decomposition; their classification is a separate and far deeper project, one strand of which appears in the simplicity of $A_n$.