Internal central products of a finite family of subgroups

Let $G$ be a group and let $G_1,\dots ,G_r$ be subgroups of $G$, where $r\in \mathbb{N}$ (Subgroup). Subgroups $G_1,\dots ,G_r$ of $G$ form an internal central product when they generate $G$ and $[G_i,G_j]=1$ for $i\mathrm{\ne }j$, where $⟨\cdot ⟩$ is the generated subgroup of The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups and $[G_i,G_j]$ is the subgroup commutator of Subgroup commutators and the lower central series. The empty family is an internal central product of the trivial group.

Since the factors commute pairwise, $G=G_1{G}_2\cdots G_r$ as a set of products, and for $i\mathrm{\ne }j$ every element of $G_i\cap G_j$ commutes with both $G_i$ and $G_j$, so $G_i\cap G_j\le Z(G_i)\cap Z(G_j)$ (The center $Z(G)$ of a group).

The condition differs from that of an internal direct product (Internal direct products of finitely many normal subgroups) in exactly one place: there the factors are required to intersect trivially, here they are allowed to share a central subgroup. An internal direct product of normal subgroups is in particular an internal central product, since distinct factors of a direct product commute elementwise.

No hypothesis is placed on the intersections beyond what the commuting condition already forces. That is deliberate: the intersections are what the recognition theorem computes, rather than data prescribed in advance.