Normal series of a group (awaiting review)

If $G$ is a group, a subnormal series for $G$ is a chain of subgroups \[ \langle e\rangle =H_0 \lhd H_1 \lhd \cdots \lhd H_k=G\] where each subgroup $H_i$ is normal in $H_{i+1}$ for all $i$.

A subnormal series where $H_i$ is normal in $G$ for all $i$ is a normal series.