principal series

《Lecture Notes on Bayesian Estimation and Classification》 · 第 63—65 页 · Section 3.1「Series and Composition Series」，群列定义及小节「Refinements」「Composition Series」

Let $G$ be an operator group with operator domain $\Omega$. An $\Omega$-series (of finite length) in $G$ is a finite sequence of $\Omega$-subgroups including $1$ and $G$ such that each member of the sequence is a normal subgroup of its successor: thus a series can be written $1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_l=G$. […] Consider the set of all $\Omega$-series of an $\Omega$-group $G$: there will always be at least one such series, namely $1\triangleleft G$. If $\mathbf S$ and $\mathbf T$ are $\Omega$-series of $G$, call $\mathbf S$ a refinement of $\mathbf T$ if every term of $\mathbf T$ is also a term of $\mathbf S$. If there is at least one term of $\mathbf S$ which is not a term of $\mathbf T$, then $\mathbf S$ is a proper refinement of $\mathbf T$. […] An $\Omega$-series which has no proper refinements is called an $\Omega$-composition series. […] If $\Omega$ is empty, we speak of a composition series. When $\Omega=\operatorname{Inn}G$, $\operatorname{Aut}G$, or $\operatorname{End}G$, an $\Omega$-composition series is called a principal series, a principal characteristic series, or a principal fully-invariant series respectively.