Upper Central Series: Doubts in Construction & Subgroups

[Pratibha]

I ve doubt in construction of upper central series. I am not sure about the subgroups appearing in upper central series ,i.e.,Z0(G),Z1(G),... ,do these subgroups form center of G?

 

[jvicens]

Hello,

Thank you for reaching out with your question about the construction of upper central series. To answer your question, the subgroups appearing in the upper central series do not necessarily form the center of G.

The upper central series is a sequence of subgroups defined by the centralizers of elements in G. The first subgroup, Z0(G), is the center of G, but the subsequent subgroups, Z1(G), Z2(G), etc., are not necessarily centers. Instead, they are defined as the centralizers of elements in the previous subgroup.

The center of a group is the set of elements that commute with all other elements in the group. In contrast, the subgroups in the upper central series are defined by the elements that commute with a particular subgroup. Therefore, the subgroups in the upper central series may contain elements that do not commute with all other elements in G, and thus do not form the center.

I hope this clarifies your doubts about the construction of upper central series. If you have any further questions, please don't hesitate to ask.