Solvable group

A solvable group has a subnormal series whose factor groups are abelian.

A group is virtually solvable, or virtually soluble, when it contains a solvable group as a finite-index subgroup. This is preserved under subgroups, quotients and group extensions. The extension proof can use a finite-index characteristic soluble subgroup of the kernel, followed by centralizing the resulting finite normal kernel. Finite generation is not required for these closure properties.

Every virtually soluble group $K$ has a soluble characteristic subgroup of finite index. First take a soluble normal subgroup $K_0$ of finite index by the subgroup core construction. Choose a soluble normal subgroup $R\supseteq K_0$ maximizing its image size in the finite quotient $K/K_0$. For every soluble normal subgroup $S$, the product $RS$ is soluble, because its quotient by $R$ is a quotient of $S$. Maximality forces $S\le R$. Hence $R$ is the unique largest soluble normal subgroup and is invariant under all automorphisms. This avoids incorrectly intersecting infinitely many conjugates when proving closure under group extensions.

The derived series is defined byA group is solvable exactly when this series reaches the trivial group after finitely many steps.

A group is metabelian when its commutator subgroup is abelian, equivalently when its derived length is at most two.

A group is polycyclic when it has a subnormal series with cyclic factors. Every subgroup and quotient of a polycyclic group is finitely generated, and extensions of polycyclic groups are polycyclic.

A group is nilpotent when it has a finite central series, equivalently when its lower central series reaches the identity.

The nilpotency class of a nilpotent group $G$ is the least nonnegative integer $s$ for which ${\gamma }_{s+1}(G)=\{1\}$ in the lower central series. Thus nontrivial abelian groups have class one.

A group $G$ is two-step nilpotent whenEquivalently, its commutator subgroup is contained in the center of a group of $G$. This allows every word to be collected into powers of generators followed by powers of their pairwise commutators.

A group is virtually nilpotent when it has a nilpotent subgroup of finite index.

The lower central series of a group is defined by ${\gamma }_1(G)=G$ and ${\gamma }_{i+1}(G)=[{\gamma }_i(G),G]$. A group is nilpotent of class at most $c$ exactly when ${\gamma }_{c+1}(G)$ is trivial.

Let $G$ be a nilpotent group of class at most $s$ in which every element has order at most $r$. A subgroup generated by $k$ elements has order at mostIndeed, collection by the lower central series expresses every element as an ordered product of simple group commutators in the generators of weights at most $s$. There are at most $k+k^2+\cdots +k^s\le s{k}^s$ such commutators, and each exponent may be reduced to one of at most $r$ values.

The free nilpotent group of rank $r$ and class $c$ is $F_r/{\gamma }_{c+1}(F_r)$, where $F_r$ is the free group of rank $r$ and ${\gamma }_i$ is its lower central series.

The quotient mapis a central nonsplit extension of the Integer Heisenberg group by ${\gamma }_3(F_2)/{\gamma }_4(F_2)\cong {\mathbb{Z}}^2$. It cannot split because the source has abelianization ${\mathbb{Z}}^2$, whereas a central splitting would give an abelianization containing an additional direct factor ${\mathbb{Z}}^2$.

A central series is a chain of normal subgroups whose successive quotients lie in the centers of the corresponding quotient groups.