Definition:Normal Series

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Definition

Let $G$ be a group whose identity is $e$.


A normal series for $G$ is a sequence of (normal) subgroups of $G$:

$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

where $G_{i - 1} \lhd G_i$ denotes that $G_{i - 1}$ is a proper normal subgroup of $G_i$.


Factors

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$


The factor groups of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$:

$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

are the quotient groups:

$G_1 / G_0, G_2 / G_1, \ldots, G_i / G_{i - 1}, \ldots, G_n / G_{n-1}$


Normal Series as Sequence of Homomorphisms

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$

whose factor groups are:

$H_1 = G_1 / G_0, H_2 = G_2 / G_1, \ldots, H_i = G_i / G_{i - 1}, \ldots, H_n = G_n / G_{n - 1}$


By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\phi_1, \ldots, \phi_n$ of group homomorphisms:

$\set e \stackrel {\phi_1} {\to} H_1 \stackrel {\phi_2} {\to} H_2 \stackrel {\phi_3} {\to} \cdots \stackrel {\phi_n} {\to} H_n$


Infinite Normal Series

A normal series may or may not terminate at either end:

$\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$

Such a series is referred to as an infinite normal series.

The context will determine which end, if either, it terminates.


Length

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}$


The length of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is the number of (normal) subgroups which make it.

In this context, the length of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is $n$.


If such a normal series is infinite, then its length is not defined.


Also known as

A normal series is also referred to as:

  • A normal tower
  • A subinvariant series
  • A sequence of groups


Also defined as

Note that from Normality Relation is not Transitive, it is not necessarily the case that if $G_a \lhd G_b \lhd G_c$ then $G_a \lhd G_c$.

Consequently, some sources specify that all subgroups $G_i$ in a normal series $\sequence {G_i}_{i \in \set {0, 1, \ldots, n} }$ be normal subgroups of $G$ itself, as well as being normal subgroups of the next in sequence $G_{i + 1}$.

Such a sequence in which it is not necessarily the case where $G_i$ is normal in $G$ for all $i$ is, in such a context, referred to as a subnormal series.


Also see

  • Results about normal series can be found here.


Sources