I write $|X|$ for the number of elements in a finite set $X$. Recall some basic facts:
If $p$ is a prime number, then any group $G$ of order $p^2$ is abelian.
Sketch of proof: Fix a prime $p$ and such a group $G$. The center $Z(G)$ of $G$ must be nontrivial by the class equation, and hence have order $p$ or $p^2$. So $G/Z(G)$ has order $p$ or $1$. So $G/Z(G)$ is cyclic. So $G$ is abelian.
If $p < q$ are prime numbers, then there is a nonabelian group of order $pq$ if and only if $q \equiv 1 \pmod{p}$, and in this case there is only one such nonabelian group (up to isomorphism).
Sketch of proof: Fix primes $p < q$ and a group $G$ of order $pq$. Use Sylow theory to conclude that $G$ has a unique (and hence normal) subgroup $H_q$ of order $q$, and a subgroup $H_p$ of order $p$, and use elementary observations to verify that $H_q \cap H_p = \{1\}$ and $G = H_q H_p$ so that $G$ is some semidirect product $H_q \rtimes H_p$. Since $H_q$ is cyclic, $|\operatorname{Aut}(H_q)| = q-1$, and so when $q \not\equiv 1 \pmod{p}$, Lagrange's theorem shows that there are no nontrivial homomorphisms $H_p \to \operatorname{Aut}(H_q)$, implying that $G$ is abelian in this case. When $q = 1 \pmod{p}$, the existence of a nontrivial homomorphisms $H_p \to \operatorname{Aut}(H_q)$ and the independence of the structure of the resulting semidirect product on the choice of nontrivial homomorphism are discussed in this math.SE post.
Together, these facts imply:
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There are no nonabelian groups of odd order less than $21$.
(The only odd numbers less than $21$ besides $1$ and primes to worry about are $9$ and $15$, which are covered by the above general results.)
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There is a unique nonabelian group of order $21$, and it can be realized by choosing any nontrivial homomorphism $\mathbb{Z}/3\mathbb{Z} \to \operatorname{Aut}(\mathbb{Z}/7\mathbb{Z})$ and forming the corresponding semidirect product. For example (taken from the above math.SE post), one could use the unique homomorphism sending $1$ to "multiplication by $2$".)
Now for my slightly subjective question:
Are there "less technical" or "more insightful" constructions of the nonabelian group of order $21$?
To be clearer about my kind of fuzzy requirements:
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I don't care if a construction doesn't shed any light on the nonexistence of nonabelian groups of smaller order.
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I do want it to be possible to define the group, and see that it is nonabelian, and of order 21, without drawing on too much group theoretical machinery.
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Ideally, the construction should minimize computation. One could, for example, exhibit two permutations of a small finite set that generate the group— but verifying that the group they generate has 21 elements and is nonabelian is entirely computational. And even if the computations are well organized, this approach gives little idea of where such a group might "come from". (If a specific presentation came with a geometrical insight making clear why the resulting group would have order $21$ without computation, that would be awesome.)
To explain the motivation behind my question: I was teaching algebra, and was asked in office hours if nonabelian groups of odd order even existed. "Of course they do," I said. Then I realized that it was rather difficult to exhibit one, given where we were in the class. We had the basic homomorphism theorems, but:
- We hadn't done the Sylow theorems or semidirect products.
- We didn't know about finite fields yet.
- All of the members of the "families of groups" we had discussed up to that point— symmetric groups, alternating groups, dihedral groups— when nonabelian, have even order.
- All nonabelian groups of order less than $21$, whether or not they into the families just mentioned, have even order.
FWIW: the "most elementary" example of a small nonabelian group of odd order that I can think of is the set of all $3 \times 3$ matrices with entries in the field with $3$ elements that are both (a) upper triangular and (b) have all $1$s down the diagonal. (This is pretty clearly a nonabelian group, and just as clearly has $27$ elements.) This happens to be the next smallest nonabelian group of odd order.
Best Answer
If $F$ is any field, there is a subgroup of $\text{PGL}_2(F)$ given by fractional linear transformation of the form $z \mapsto az + b$ where $a \in F^{\times}, b \in F$. This is precisely the subgroup fixing the point at infinity in $\mathbb{P}^1(F)$. Taking $F$ to be a finite field $\mathbb{F}_q$ we obtain a family of (usually) nonabelian Frobenius groups of order $q(q - 1)$.
Now, we can further restrict $a$ to lie in any subgroup of $F^{\times}$; in particular, for $q$ odd, taking it to lie in the subgroup of quadratic residues gives a family of (usually) nonabelian groups of order $\frac{q(q-1)}{2}$. Taking $q = 7$ gives the desired group.