Nilpotent group

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

For $tv{t}^{-1}=Bv$, the commutator $[t,v]$ is $(B-I)v$ in additive notation. The lower central series from its second term is $(B-I){\mathbb{Z}}^r,(B-I{)}^2{\mathbb{Z}}^r,\dots$. It terminates exactly when $B-I$ is a nilpotent matrix. For rank two this means $(B-I{)}^2=0$, equivalently $\det B=1$ and $\mathrm{tr}B=2$. Nilpotence of a finite-index subgroup alone gives virtual nilpotence of the whole group.

Every finitely generated torsion-free group which is a nilpotent group embeds as a discrete cocompact subgroup of a unique simply connected nilpotent real Lie group. Its dimension equals the group's integral cohomological dimension of a group. In dimension three the Lie algebra is either abelian or has a basis with a single nonzero bracket $[Y,T]=X$, giving the real Heisenberg group.

If a nilpotent group is generated by $a_1,\dots ,a_d$, every element of its commutator subgroup is a product $[x_1,a_1]\cdots [x_d,a_d]$. The sets in the displayed formula need not be subgroups. Induct on the nilpotency class $c$. Collection in $G/{\gamma }_c(G)$ gives the desired product with a central error. Since ${\gamma }_c(G)=[{\gamma }_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots [y_d,a_d]$ with $y_i\in {\gamma }_{c-1}(G)$. These factors are central, and ${\gamma }_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_i{y}_i$ absorbs the error. For a finitely generated pro-p group, applying this formula in all finite quotients and using compactness makes its algebraically generated commutator subgroup closed.

If a nilpotent group is generated by $a_1,\dots ,a_d$, every element of its commutator subgroup is a product $[x_1,a_1]\cdots [x_d,a_d]$. The sets in the displayed formula need not be subgroups. Induct on the nilpotency class $c$. Collection in $G/{\gamma }_c(G)$ gives the desired product with a central error. Since ${\gamma }_c(G)=[{\gamma }_{c-1}(G),G]$, that error is a product $[y_1,a_1]\cdots [y_d,a_d]$ with $y_i\in {\gamma }_{c-1}(G)$. These factors are central, and ${\gamma }_{c-1}(G)$ centralizes $G\prime$, so replacing $x_i$ by $x_i{y}_i$ absorbs the error. For a finitely generated pro-p group, applying this formula in all finite quotients and using compactness makes its algebraically generated commutator subgroup closed.

A finite group is nilpotent exactly when it is the direct product of groups of its Sylow subgroups. The normalizer condition for nilpotent groups and Sylow conjugacy force each Sylow normalizer to be the whole group. Distinct normal Sylow subgroups commute because their commutators lie in their trivial intersection. Conversely finite p-groups are nilpotent by the nontrivial center and induction; direct products preserve nilpotence.

Choose the first term of the upper central series not contained in $H$. The preceding term lies in $H$, so an element of the new term outside $H$ commutes with $H$ modulo that preceding term and normalizes $H$.

Repeatedly take the center of a group after quotienting by the preceding term. A group is a nilpotent group exactly when a finite term is the whole group. This yields the normalizer condition for nilpotent groups.

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.

Group commutators are taken coordinatewise in a direct product of groups, giving ${\Gamma }_j(A\times B)={\Gamma }_j(A)\times {\Gamma }_j(B)$. A finite direct product of nilpotent groups is nilpotent, with class the maximum of the factor classes.

// Target: algebra.bigb

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.

In a two-generated two-step nilpotent group, use $c=[x,y]=x^{-1}{y}^{-1}xy$. Centrality of $c$ gives $yx=xy{c}^{-1}$, so every word collects to $x^a{y}^b{c}^r$. The multiplication of these collected expressions is $(a,b,r)(a^{\prime },b^{\prime },r^{\prime })=(a+a^{\prime },b+b^{\prime },r+r^{\prime }-a^{\prime }b)$. Relations on the central commutator can reduce $r$ modulo its order. To turn collection into a word problem for a group algorithm, one must also prove the independence of the remaining coordinates, for example by constructing a group with that cocycle multiplication.

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.

For a length-$r$ central series $1=G_0\le \cdots \le G_r=G$, induction from the top gives ${\Gamma }_{r-i+1}(G)\le G_i$, and induction from the bottom gives $G_i\le Z_i(G)$. Consequently the lower central series terminates at ${\Gamma }_{c+1}=1$ exactly when the upper central series reaches $Z_c=G$.

// Target: algebra.bigb