Definition:Normal Series/Also defined as

Normal Series: 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 $\sequence {G_i}_{i \mathop \in \set {0, 1, \ldots, n} }$ 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.