Let $G$ be a group with a subgroup $H$ , and let \begin{equation} G = G_0 \rhd G_1 \rhd \cdots \rhd G_n = H \end{equation}be a series of subgroups with each $G_i$ a normal subgroup of $G_{i-1}$ . Such a series is called a subnormal series or a subinvariant series.

If in addition, each $G_i$ is a normal subgroup of $G$ , then the series is called a normal series.

A subnormal series in which each $G_i$ is a maximal normal subgroup of $G_{i-1}$ is called a composition series.

A normal series in which $G_i$ is a maximal normal subgroup of $G$ contained in $G_{i-1}$ is called a principal series or a chief series.

Note that a composition series need not end in the trivial group $1$ . One speaks of a composition series (1) as a composition series from $G$ to $H$ . But the term composition series for $G$ generally means a composition series from $G$ to $1$ .

Similar remarks apply to principal series.

Some authors use normal series as a synonym for subnormal series. This usage is, of course, not compatible with the stronger definition of normal series given above.