Definition:Normal Series/Sequence of Homomorphisms

Definition
Let $G$ be a group whose identity is $e$. Let $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]}$ be a normal series for $G$:
 * $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]} = \left({\left\{{e}\right\} = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}\right)$

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:


 * $\left\{{e}\right\} \stackrel {\phi_1}{\to} H_1 \stackrel {\phi_2}{\to} H_2 \stackrel {\phi_3}{\to} \cdots \stackrel {\phi_n}{\to} H_n$