In the mathematical field of group theory, a central series is a sequence of normal subgroups of a group that satisfies specific properties related to the center of the group or its quotients. These series are fundamental tools for understanding the structure of groups, particularly nilpotent groups, and provide a measure of how "non-abelian" a group is. There are two primary types: the descending central series (also known as the lower central series) and the ascending central series (also known as the upper central series).
Descending Central Series (Lower Central Series)
The descending central series of a group $G$, denoted by $\gamma_1(G), \gamma_2(G), \gamma_3(G), \dots$, is a sequence of normal subgroups defined recursively as follows:
- $\gamma_1(G) = G$
- $\gamma_{n+1}(G) = [\gamma_n(G), G]$ for $n \ge 1$. Here, $[\gamma_n(G), G]$ represents the commutator subgroup generated by all commutators $[x, y] = x^{-1}y^{-1}xy$ where $x \in \gamma_n(G)$ and $y \in G$.
This definition yields a chain of normal subgroups: $G = \gamma_1(G) \supseteq \gamma_2(G) \supseteq \gamma_3(G) \supseteq \dots$
The term $\gamma_2(G) = [G, G]$ is the derived subgroup (or commutator subgroup) of $G$. Each subsequent term $\gamma_{n+1}(G)$ consists of "higher-order commutators." An important property is that each quotient group $\gamma_n(G)/\gamma_{n+1}(G)$ is abelian.
A group $G$ is defined as nilpotent if its descending central series terminates at the trivial subgroup ${e}$ in a finite number of steps. That is, there exists an integer $c$ such that $\gamma_c(G) = {e}$. The smallest such integer $c$ is called the nilpotency class of $G$.
Ascending Central Series (Upper Central Series)
The ascending central series of a group $G$, denoted by $Z_0(G), Z_1(G), Z_2(G), \dots$, is a sequence of normal subgroups defined recursively as follows:
- $Z_0(G) = {e}$, the trivial subgroup.
- $Z_{n+1}(G)$ is the subgroup of $G$ such that $Z_{n+1}(G)/Z_n(G)$ is the center of the quotient group $G/Z_n(G)$. Formally, $Z_{n+1}(G)/Z_n(G) = Z(G/Z_n(G))$.
This definition yields an ascending chain of normal subgroups: ${e} = Z_0(G) \subseteq Z_1(G) \subseteq Z_2(G) \subseteq \dots$
The first term $Z_1(G)$ is the center of $G$, denoted $Z(G)$. Each subsequent term $Z_{n+1}(G)$ consists of all elements $g \in G$ such that for every element $x \in G$, the commutator $[g, x]$ belongs to $Z_n(G)$.
A group $G$ is defined as nilpotent if its ascending central series terminates at the group $G$ itself in a finite number of steps. That is, there exists an integer $c$ such that $Z_c(G) = G$. The smallest such integer $c$ is called the nilpotency class of $G$. Both the descending and ascending central series provide equivalent definitions for nilpotency, and the nilpotency class derived from either series is the same.
Significance and Properties
- Nilpotency: Central series are the defining characteristic of nilpotent groups. Nilpotent groups generalize abelian groups; an abelian group is a nilpotent group of class 1 (since $Z(G)=G$ and $\gamma_2(G)={e}$).
- Group Structure: They provide a hierarchical way to decompose and analyze the structure of a group, indicating how far a group deviates from being abelian.
- Relationship between Series: For a nilpotent group of class $c$, the ascending and descending central series are closely related. Specifically, for each $k \in {0, \dots, c}$, $Z_k(G)$ is the largest subgroup $H$ such that $\gamma_{c-k+1}(G) \subseteq H$.
- Burnside Basis Theorem: For finite $p$-groups, the lower central series is linked to the Frattini subgroup and provides insights into the minimal generating sets of the group.
- Generalizations: The concept of central series is not limited to groups; analogous notions of lower central series and nilpotency exist in other algebraic structures, such as Lie algebras.