Frobenius group: Z7⋊Z3

The group Z7⋊Z3 is the smallest nonabelian group of odd order. It is a group of order 21.

The group is the semidirect product of Z7 and Z3.

This group is soluble group.

Construction as a semidirect product

We will construct this group as a semidirect product ${\mathbb{Z}}_7⋊{\mathbb{Z}}_3$.

We need to find a group homomorphism $\rho :{\mathbb{Z}}_3\to \mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{Z}}_7)\cong {\mathbb{Z}}_6$.

Consider the map $x\mapsto (y\mapsto 2^{x}y)$

Here, $x$ is an element of the integers mod $3$, and $y$ is an element of the integers mod $7$.

This is a homomorphism, since $\rho (x{x}^{\prime })=y\mapsto 2^{x{x}^{\prime }}y=(y\mapsto 2^{x}y)\circ (y\mapsto 2^{x^{\prime }}y)=\rho (x)\rho (x^{\prime })$. Indeed, $\rho (1)=y\mapsto y$, the identity element of $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{Z}}_7)$.

Then $G={\mathbb{Z}}_7{⋊}_{\rho }{\mathbb{Z}}_3$ is non-abelian since $(0,1)*(1,0)=(2,1)$, $(1,0)*(0,1)=(1,1)$.

Hence we have constructed a non-abelian group of order $21$.

The classification of groups of order 21 says that there is only one non-abelian group of order $21$, the Frobenius group, hence this is the Frobenius group.