Presentation of a finite group (awaiting review)

A presentation of a group $G$ is a description of $G$ as the quotient $F/R$ of a free group $F$ generated by a specified set of generators, modulo the normal subgroup $R$ generated by a set of words in those generators. When $G$ is abelian we instead express $G$ as a quotient of a free abelian group $F$ so that we can omit commutator relations.

In what follows, we denote by $g^h$ the conjugate $h^{-1}gh$ and by $[g, h]$ the commutator $ghg^{-1}h^{-1}$.

We only give presentations for finite solvable groups, where they can take a special form. A polycyclic series is a subnormal series $G = G_1 \trianglerighteq G_2 \trianglerighteq \dots \trianglerighteq G_n \trianglerighteq G_{n+1} = \{1\}$ so that $G_i/G_{i+1}$ is cyclic for each $i$. A polycyclic sequence is a sequence of elements $(g_1, \dots, g_n)$ of $G$ so that $G_i/G_{i+1} = \langle g_i G_{i+1}\rangle$. The relative orders of a polycyclic series are the orders $r_i$ of the cyclic quotients $G_i / G_{i+1}$. The polycyclic presentation associated to a polycyclic sequence has generators $g_1, \dots, g_n$ and relations of the following shape.

• $g_i^{r_i} = \prod_{k=i+1}^n g_k^{a_{i,k}}$ for all $i$;
• $g_i^{g_j} = \prod_{k=j+1}^n g_k^{b_{i,j,k}}$ for $j < i$.

Any finite solvable group has a polycyclic presentation. When the size of $G$ is not too large, we choose a presentation with the following properties:

• it has a minimal number of generators;
• among such, it has a maximal number of $i$ so that all $a_{i,k} = 0$;
• among such, it has a maximal number of commuting $g_i$;
• among such, aim for an increasing sequence of relative orders;
• among such, minimize the sum of the $b_{i,j,k}$ for noncommuting generators $g_i$ and $g_j$.