Definition:Normal Series/Sequence of Homomorphisms

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Definition

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

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:

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


Sources