Affine General Linear Group (AGL)

Specifically focusing on one-dimensional affine general linear groups $AGL(1, n)$, this group consists of the linear transformations $ax + b$, where $a$ and $b$ are taken mod $n$, and $gcd(a, n) = 1$ ($a$ is relatively prime to $n$), with the group operation being composition of these linear transformations. This group is useful to group cipher cryptography because in terms of permutations of a ciphertext alphabet of length $n$, it consists of all rotations and decimations of the alphabet. This is a non-commutative group (assuming $n > 2$) with $\varphi(n) (n-1)$ elements. $AGL(1, 3)$ is isomorphic to $S_3$ (a symmetric group). The group can be factorized as a semidirect product of the cyclic group $C_n$ and the group of units $(C_n)^x$. This group of units is the multiplicative group of numbers mod $n$ which are relatively prime to $n$, and is commutative with a polycyclic structure. For prime $n$, $(C_n)^x$ is just $C_{n-1}$.